Largest multiple of $7$ lower than some $78$-digit number?












5












$begingroup$


What I am trying to achieve, is related to cryptography/blockchain/bitcoin . So, the largest number here is huge, in other words: I want to find the largest multiple of 7, which is lower than this number:



$115792089237316195423570985008687907852837564279074904382605163141518161494336 $



I can just go to Wolfram Alpha, and type "multiples of 7", and I get a list of the multiples relatively fast. But, it will take some time until I keep hitting "more", to get to a number lower than this above.










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  • $begingroup$
    m.wolframalpha.com/input/… works a lot easier it lists as 2 mod 7.
    $endgroup$
    – Roddy MacPhee
    17 hours ago










  • $begingroup$
    78 digits is semi small cryptographically compared to 617 digit (2048 bit) crypto keys.
    $endgroup$
    – Roddy MacPhee
    17 hours ago






  • 1




    $begingroup$
    I would very much like (aka 'prefer') to see a solution which does not require use of extended-precision or large-integer software packages. (Compare with, e.g., the simple "sum the digits" approach for multiples of 3). Is Roddy's answer the only such?
    $endgroup$
    – Carl Witthoft
    17 hours ago










  • $begingroup$
    no answer requires it. it's simply more convenient for numbers of this size ( I speak from experience, thought I messed up because it didn't match the other answer, turns out I was doing the mod 7 steps too early.found that out by calculator) you can literally do mod as you would long division, just forget to write out the quotient.
    $endgroup$
    – Roddy MacPhee
    16 hours ago








  • 1




    $begingroup$
    +1 for "it will take some time"!
    $endgroup$
    – TonyK
    14 hours ago
















5












$begingroup$


What I am trying to achieve, is related to cryptography/blockchain/bitcoin . So, the largest number here is huge, in other words: I want to find the largest multiple of 7, which is lower than this number:



$115792089237316195423570985008687907852837564279074904382605163141518161494336 $



I can just go to Wolfram Alpha, and type "multiples of 7", and I get a list of the multiples relatively fast. But, it will take some time until I keep hitting "more", to get to a number lower than this above.










share|cite|improve this question









New contributor




kpopguy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    m.wolframalpha.com/input/… works a lot easier it lists as 2 mod 7.
    $endgroup$
    – Roddy MacPhee
    17 hours ago










  • $begingroup$
    78 digits is semi small cryptographically compared to 617 digit (2048 bit) crypto keys.
    $endgroup$
    – Roddy MacPhee
    17 hours ago






  • 1




    $begingroup$
    I would very much like (aka 'prefer') to see a solution which does not require use of extended-precision or large-integer software packages. (Compare with, e.g., the simple "sum the digits" approach for multiples of 3). Is Roddy's answer the only such?
    $endgroup$
    – Carl Witthoft
    17 hours ago










  • $begingroup$
    no answer requires it. it's simply more convenient for numbers of this size ( I speak from experience, thought I messed up because it didn't match the other answer, turns out I was doing the mod 7 steps too early.found that out by calculator) you can literally do mod as you would long division, just forget to write out the quotient.
    $endgroup$
    – Roddy MacPhee
    16 hours ago








  • 1




    $begingroup$
    +1 for "it will take some time"!
    $endgroup$
    – TonyK
    14 hours ago














5












5








5





$begingroup$


What I am trying to achieve, is related to cryptography/blockchain/bitcoin . So, the largest number here is huge, in other words: I want to find the largest multiple of 7, which is lower than this number:



$115792089237316195423570985008687907852837564279074904382605163141518161494336 $



I can just go to Wolfram Alpha, and type "multiples of 7", and I get a list of the multiples relatively fast. But, it will take some time until I keep hitting "more", to get to a number lower than this above.










share|cite|improve this question









New contributor




kpopguy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




What I am trying to achieve, is related to cryptography/blockchain/bitcoin . So, the largest number here is huge, in other words: I want to find the largest multiple of 7, which is lower than this number:



$115792089237316195423570985008687907852837564279074904382605163141518161494336 $



I can just go to Wolfram Alpha, and type "multiples of 7", and I get a list of the multiples relatively fast. But, it will take some time until I keep hitting "more", to get to a number lower than this above.







elementary-number-theory modular-arithmetic arithmetic






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kpopguy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









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share|cite|improve this question




share|cite|improve this question








edited 13 hours ago









user21820

39.4k543155




39.4k543155






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asked 19 hours ago









kpopguykpopguy

374




374




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  • $begingroup$
    m.wolframalpha.com/input/… works a lot easier it lists as 2 mod 7.
    $endgroup$
    – Roddy MacPhee
    17 hours ago










  • $begingroup$
    78 digits is semi small cryptographically compared to 617 digit (2048 bit) crypto keys.
    $endgroup$
    – Roddy MacPhee
    17 hours ago






  • 1




    $begingroup$
    I would very much like (aka 'prefer') to see a solution which does not require use of extended-precision or large-integer software packages. (Compare with, e.g., the simple "sum the digits" approach for multiples of 3). Is Roddy's answer the only such?
    $endgroup$
    – Carl Witthoft
    17 hours ago










  • $begingroup$
    no answer requires it. it's simply more convenient for numbers of this size ( I speak from experience, thought I messed up because it didn't match the other answer, turns out I was doing the mod 7 steps too early.found that out by calculator) you can literally do mod as you would long division, just forget to write out the quotient.
    $endgroup$
    – Roddy MacPhee
    16 hours ago








  • 1




    $begingroup$
    +1 for "it will take some time"!
    $endgroup$
    – TonyK
    14 hours ago


















  • $begingroup$
    m.wolframalpha.com/input/… works a lot easier it lists as 2 mod 7.
    $endgroup$
    – Roddy MacPhee
    17 hours ago










  • $begingroup$
    78 digits is semi small cryptographically compared to 617 digit (2048 bit) crypto keys.
    $endgroup$
    – Roddy MacPhee
    17 hours ago






  • 1




    $begingroup$
    I would very much like (aka 'prefer') to see a solution which does not require use of extended-precision or large-integer software packages. (Compare with, e.g., the simple "sum the digits" approach for multiples of 3). Is Roddy's answer the only such?
    $endgroup$
    – Carl Witthoft
    17 hours ago










  • $begingroup$
    no answer requires it. it's simply more convenient for numbers of this size ( I speak from experience, thought I messed up because it didn't match the other answer, turns out I was doing the mod 7 steps too early.found that out by calculator) you can literally do mod as you would long division, just forget to write out the quotient.
    $endgroup$
    – Roddy MacPhee
    16 hours ago








  • 1




    $begingroup$
    +1 for "it will take some time"!
    $endgroup$
    – TonyK
    14 hours ago
















$begingroup$
m.wolframalpha.com/input/… works a lot easier it lists as 2 mod 7.
$endgroup$
– Roddy MacPhee
17 hours ago




$begingroup$
m.wolframalpha.com/input/… works a lot easier it lists as 2 mod 7.
$endgroup$
– Roddy MacPhee
17 hours ago












$begingroup$
78 digits is semi small cryptographically compared to 617 digit (2048 bit) crypto keys.
$endgroup$
– Roddy MacPhee
17 hours ago




$begingroup$
78 digits is semi small cryptographically compared to 617 digit (2048 bit) crypto keys.
$endgroup$
– Roddy MacPhee
17 hours ago




1




1




$begingroup$
I would very much like (aka 'prefer') to see a solution which does not require use of extended-precision or large-integer software packages. (Compare with, e.g., the simple "sum the digits" approach for multiples of 3). Is Roddy's answer the only such?
$endgroup$
– Carl Witthoft
17 hours ago




$begingroup$
I would very much like (aka 'prefer') to see a solution which does not require use of extended-precision or large-integer software packages. (Compare with, e.g., the simple "sum the digits" approach for multiples of 3). Is Roddy's answer the only such?
$endgroup$
– Carl Witthoft
17 hours ago












$begingroup$
no answer requires it. it's simply more convenient for numbers of this size ( I speak from experience, thought I messed up because it didn't match the other answer, turns out I was doing the mod 7 steps too early.found that out by calculator) you can literally do mod as you would long division, just forget to write out the quotient.
$endgroup$
– Roddy MacPhee
16 hours ago






$begingroup$
no answer requires it. it's simply more convenient for numbers of this size ( I speak from experience, thought I messed up because it didn't match the other answer, turns out I was doing the mod 7 steps too early.found that out by calculator) you can literally do mod as you would long division, just forget to write out the quotient.
$endgroup$
– Roddy MacPhee
16 hours ago






1




1




$begingroup$
+1 for "it will take some time"!
$endgroup$
– TonyK
14 hours ago




$begingroup$
+1 for "it will take some time"!
$endgroup$
– TonyK
14 hours ago










3 Answers
3






active

oldest

votes


















13












$begingroup$

One can compute this number $a$ modulo $7$. The result is $2bmod 7$. So take $a-2$. It is the largest multiple of $7$ less than $a$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thanks, this should work
    $endgroup$
    – kpopguy
    19 hours ago










  • $begingroup$
    @PeterSzilas See How to compute modulo, or similar links. With wolframalpha, compute it here.
    $endgroup$
    – Dietrich Burde
    19 hours ago












  • $begingroup$
    Dietrich.Thanks.Any advantage to using the modulo calculator, why not just divide (calculator) and find remainder ?You have to enter this lengthy number anyway? Hope this question is not too trivial.Thanks.
    $endgroup$
    – Peter Szilas
    19 hours ago






  • 1




    $begingroup$
    @Peter When you are using such big numbers (and given the context), you would usually not be typing in the number, but writing a program (in Python or Go, etc). This is a more efficient method than, say dividing the number by 7 and the successively smaller numbers till the remainder is 0.
    $endgroup$
    – Devashish Kaushik
    18 hours ago










  • $begingroup$
    Davashish.Thank you. My questions came up when I read 2 mod 7, where from ?As you can see not a number cruncher:)Thank you for your comment.
    $endgroup$
    – Peter Szilas
    17 hours ago





















5












$begingroup$

$$begin{array}{cccccc}115792&089237&316195&423570&985008&687907\852837&564279&074904&382605&163141&518161\494336end{array}$$
Sum up the places of these numbers, by place value carrying when needed, then apply $10^kequiv 3^k bmod 7$ you'll then have a much smaller number to find the remainder of that's equivalent.
5667972, which goes to :$$6(3^5)+6(3^4)+2(3^2)equiv 1458+486+18equiv 2+3+4equiv 2 bmod 7$$ so the largest multiple of 7, is 2 less than the number. Yes, this is a slightly tedious way to go, but it's inspired via extension of Fermat's little theorem, and polynomial remainder theorem.



The reason I broke it into 6 digits at a time, is because Fermat's extension, is that exponents that have the same remainder mod p-1, will give back the same remainder with the same base. That means you can simply turn one into the other, adding like terms. you then go and do the addition the first column on the right sums to 62, carry the 6, that means you sum the next column plus 6, giving 57 carry the 5, next column is then 59, carry the 5, next column 67, carry the 6, next column, 76 carry the 7, next column, 56 there's no column to carry the 5 onto, and in the next step, it will be merged with the 2 (6 digits before), and then tossed because 7 creates a term that is 0 mod 7. Doing the same to other 7's and the nine gives 660200 we then replace x=10 with 3, via polynomial remainder theorem, and evaluate the sum shown above.Formula used $$sum_{n=0}^Ld_na^nequivsum_{n=0}^L(d_nbmod p)(a_nbmod p)^{(n bmod (p-1))} pmod p$$ we did the exponent part first, the base part second, and the coefficient (digit) part third, we then used the simple reduction mod p last.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    and had I not made an stupid error, with this small modulus, no calculator required.
    $endgroup$
    – Roddy MacPhee
    16 hours ago










  • $begingroup$
    I think this requires a bit more step-by-step explanation...
    $endgroup$
    – jcaron
    15 hours ago










  • $begingroup$
    All the steps are literally in there. but I guess I can show the summation instead of talking about it.
    $endgroup$
    – Roddy MacPhee
    15 hours ago










  • $begingroup$
    It also has a remainder of 2 if evaluated as hexadecimal.
    $endgroup$
    – Roddy MacPhee
    10 hours ago










  • $begingroup$
    Ouch! Your added explanation has confused me more than the original :-( . Maybe you could identify exactly what "first column on the right" is, and what exactly is summing to 62 (sum of the modulos?) . And so on. Or are you calculating , e.g., 115792 mod 7 and adding that to the sum of 10^k for that group?
    $endgroup$
    – Carl Witthoft
    9 hours ago



















1












$begingroup$

Just divide the number by 7, if the mod is 0, you subtract 1 from the quotient and multiply it by 7, else, the quotient times 7 is your desired number.



Ex:
70 / 7 = 10, with mod 0. 10-1 = 9 => 9 * 7 = 63 >Biggest multiple under 70.



71 / 7 = 10, with mod 1. 10 * 7 = 70 => Biggest multiple under 71






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New contributor




Fabio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$













  • $begingroup$
    That's only fast to a point, and only if you know your multiples.
    $endgroup$
    – Roddy MacPhee
    14 hours ago











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3 Answers
3






active

oldest

votes








3 Answers
3






active

oldest

votes









active

oldest

votes






active

oldest

votes









13












$begingroup$

One can compute this number $a$ modulo $7$. The result is $2bmod 7$. So take $a-2$. It is the largest multiple of $7$ less than $a$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thanks, this should work
    $endgroup$
    – kpopguy
    19 hours ago










  • $begingroup$
    @PeterSzilas See How to compute modulo, or similar links. With wolframalpha, compute it here.
    $endgroup$
    – Dietrich Burde
    19 hours ago












  • $begingroup$
    Dietrich.Thanks.Any advantage to using the modulo calculator, why not just divide (calculator) and find remainder ?You have to enter this lengthy number anyway? Hope this question is not too trivial.Thanks.
    $endgroup$
    – Peter Szilas
    19 hours ago






  • 1




    $begingroup$
    @Peter When you are using such big numbers (and given the context), you would usually not be typing in the number, but writing a program (in Python or Go, etc). This is a more efficient method than, say dividing the number by 7 and the successively smaller numbers till the remainder is 0.
    $endgroup$
    – Devashish Kaushik
    18 hours ago










  • $begingroup$
    Davashish.Thank you. My questions came up when I read 2 mod 7, where from ?As you can see not a number cruncher:)Thank you for your comment.
    $endgroup$
    – Peter Szilas
    17 hours ago


















13












$begingroup$

One can compute this number $a$ modulo $7$. The result is $2bmod 7$. So take $a-2$. It is the largest multiple of $7$ less than $a$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thanks, this should work
    $endgroup$
    – kpopguy
    19 hours ago










  • $begingroup$
    @PeterSzilas See How to compute modulo, or similar links. With wolframalpha, compute it here.
    $endgroup$
    – Dietrich Burde
    19 hours ago












  • $begingroup$
    Dietrich.Thanks.Any advantage to using the modulo calculator, why not just divide (calculator) and find remainder ?You have to enter this lengthy number anyway? Hope this question is not too trivial.Thanks.
    $endgroup$
    – Peter Szilas
    19 hours ago






  • 1




    $begingroup$
    @Peter When you are using such big numbers (and given the context), you would usually not be typing in the number, but writing a program (in Python or Go, etc). This is a more efficient method than, say dividing the number by 7 and the successively smaller numbers till the remainder is 0.
    $endgroup$
    – Devashish Kaushik
    18 hours ago










  • $begingroup$
    Davashish.Thank you. My questions came up when I read 2 mod 7, where from ?As you can see not a number cruncher:)Thank you for your comment.
    $endgroup$
    – Peter Szilas
    17 hours ago
















13












13








13





$begingroup$

One can compute this number $a$ modulo $7$. The result is $2bmod 7$. So take $a-2$. It is the largest multiple of $7$ less than $a$.






share|cite|improve this answer









$endgroup$



One can compute this number $a$ modulo $7$. The result is $2bmod 7$. So take $a-2$. It is the largest multiple of $7$ less than $a$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 19 hours ago









Dietrich BurdeDietrich Burde

81k648106




81k648106












  • $begingroup$
    thanks, this should work
    $endgroup$
    – kpopguy
    19 hours ago










  • $begingroup$
    @PeterSzilas See How to compute modulo, or similar links. With wolframalpha, compute it here.
    $endgroup$
    – Dietrich Burde
    19 hours ago












  • $begingroup$
    Dietrich.Thanks.Any advantage to using the modulo calculator, why not just divide (calculator) and find remainder ?You have to enter this lengthy number anyway? Hope this question is not too trivial.Thanks.
    $endgroup$
    – Peter Szilas
    19 hours ago






  • 1




    $begingroup$
    @Peter When you are using such big numbers (and given the context), you would usually not be typing in the number, but writing a program (in Python or Go, etc). This is a more efficient method than, say dividing the number by 7 and the successively smaller numbers till the remainder is 0.
    $endgroup$
    – Devashish Kaushik
    18 hours ago










  • $begingroup$
    Davashish.Thank you. My questions came up when I read 2 mod 7, where from ?As you can see not a number cruncher:)Thank you for your comment.
    $endgroup$
    – Peter Szilas
    17 hours ago




















  • $begingroup$
    thanks, this should work
    $endgroup$
    – kpopguy
    19 hours ago










  • $begingroup$
    @PeterSzilas See How to compute modulo, or similar links. With wolframalpha, compute it here.
    $endgroup$
    – Dietrich Burde
    19 hours ago












  • $begingroup$
    Dietrich.Thanks.Any advantage to using the modulo calculator, why not just divide (calculator) and find remainder ?You have to enter this lengthy number anyway? Hope this question is not too trivial.Thanks.
    $endgroup$
    – Peter Szilas
    19 hours ago






  • 1




    $begingroup$
    @Peter When you are using such big numbers (and given the context), you would usually not be typing in the number, but writing a program (in Python or Go, etc). This is a more efficient method than, say dividing the number by 7 and the successively smaller numbers till the remainder is 0.
    $endgroup$
    – Devashish Kaushik
    18 hours ago










  • $begingroup$
    Davashish.Thank you. My questions came up when I read 2 mod 7, where from ?As you can see not a number cruncher:)Thank you for your comment.
    $endgroup$
    – Peter Szilas
    17 hours ago


















$begingroup$
thanks, this should work
$endgroup$
– kpopguy
19 hours ago




$begingroup$
thanks, this should work
$endgroup$
– kpopguy
19 hours ago












$begingroup$
@PeterSzilas See How to compute modulo, or similar links. With wolframalpha, compute it here.
$endgroup$
– Dietrich Burde
19 hours ago






$begingroup$
@PeterSzilas See How to compute modulo, or similar links. With wolframalpha, compute it here.
$endgroup$
– Dietrich Burde
19 hours ago














$begingroup$
Dietrich.Thanks.Any advantage to using the modulo calculator, why not just divide (calculator) and find remainder ?You have to enter this lengthy number anyway? Hope this question is not too trivial.Thanks.
$endgroup$
– Peter Szilas
19 hours ago




$begingroup$
Dietrich.Thanks.Any advantage to using the modulo calculator, why not just divide (calculator) and find remainder ?You have to enter this lengthy number anyway? Hope this question is not too trivial.Thanks.
$endgroup$
– Peter Szilas
19 hours ago




1




1




$begingroup$
@Peter When you are using such big numbers (and given the context), you would usually not be typing in the number, but writing a program (in Python or Go, etc). This is a more efficient method than, say dividing the number by 7 and the successively smaller numbers till the remainder is 0.
$endgroup$
– Devashish Kaushik
18 hours ago




$begingroup$
@Peter When you are using such big numbers (and given the context), you would usually not be typing in the number, but writing a program (in Python or Go, etc). This is a more efficient method than, say dividing the number by 7 and the successively smaller numbers till the remainder is 0.
$endgroup$
– Devashish Kaushik
18 hours ago












$begingroup$
Davashish.Thank you. My questions came up when I read 2 mod 7, where from ?As you can see not a number cruncher:)Thank you for your comment.
$endgroup$
– Peter Szilas
17 hours ago






$begingroup$
Davashish.Thank you. My questions came up when I read 2 mod 7, where from ?As you can see not a number cruncher:)Thank you for your comment.
$endgroup$
– Peter Szilas
17 hours ago













5












$begingroup$

$$begin{array}{cccccc}115792&089237&316195&423570&985008&687907\852837&564279&074904&382605&163141&518161\494336end{array}$$
Sum up the places of these numbers, by place value carrying when needed, then apply $10^kequiv 3^k bmod 7$ you'll then have a much smaller number to find the remainder of that's equivalent.
5667972, which goes to :$$6(3^5)+6(3^4)+2(3^2)equiv 1458+486+18equiv 2+3+4equiv 2 bmod 7$$ so the largest multiple of 7, is 2 less than the number. Yes, this is a slightly tedious way to go, but it's inspired via extension of Fermat's little theorem, and polynomial remainder theorem.



The reason I broke it into 6 digits at a time, is because Fermat's extension, is that exponents that have the same remainder mod p-1, will give back the same remainder with the same base. That means you can simply turn one into the other, adding like terms. you then go and do the addition the first column on the right sums to 62, carry the 6, that means you sum the next column plus 6, giving 57 carry the 5, next column is then 59, carry the 5, next column 67, carry the 6, next column, 76 carry the 7, next column, 56 there's no column to carry the 5 onto, and in the next step, it will be merged with the 2 (6 digits before), and then tossed because 7 creates a term that is 0 mod 7. Doing the same to other 7's and the nine gives 660200 we then replace x=10 with 3, via polynomial remainder theorem, and evaluate the sum shown above.Formula used $$sum_{n=0}^Ld_na^nequivsum_{n=0}^L(d_nbmod p)(a_nbmod p)^{(n bmod (p-1))} pmod p$$ we did the exponent part first, the base part second, and the coefficient (digit) part third, we then used the simple reduction mod p last.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    and had I not made an stupid error, with this small modulus, no calculator required.
    $endgroup$
    – Roddy MacPhee
    16 hours ago










  • $begingroup$
    I think this requires a bit more step-by-step explanation...
    $endgroup$
    – jcaron
    15 hours ago










  • $begingroup$
    All the steps are literally in there. but I guess I can show the summation instead of talking about it.
    $endgroup$
    – Roddy MacPhee
    15 hours ago










  • $begingroup$
    It also has a remainder of 2 if evaluated as hexadecimal.
    $endgroup$
    – Roddy MacPhee
    10 hours ago










  • $begingroup$
    Ouch! Your added explanation has confused me more than the original :-( . Maybe you could identify exactly what "first column on the right" is, and what exactly is summing to 62 (sum of the modulos?) . And so on. Or are you calculating , e.g., 115792 mod 7 and adding that to the sum of 10^k for that group?
    $endgroup$
    – Carl Witthoft
    9 hours ago
















5












$begingroup$

$$begin{array}{cccccc}115792&089237&316195&423570&985008&687907\852837&564279&074904&382605&163141&518161\494336end{array}$$
Sum up the places of these numbers, by place value carrying when needed, then apply $10^kequiv 3^k bmod 7$ you'll then have a much smaller number to find the remainder of that's equivalent.
5667972, which goes to :$$6(3^5)+6(3^4)+2(3^2)equiv 1458+486+18equiv 2+3+4equiv 2 bmod 7$$ so the largest multiple of 7, is 2 less than the number. Yes, this is a slightly tedious way to go, but it's inspired via extension of Fermat's little theorem, and polynomial remainder theorem.



The reason I broke it into 6 digits at a time, is because Fermat's extension, is that exponents that have the same remainder mod p-1, will give back the same remainder with the same base. That means you can simply turn one into the other, adding like terms. you then go and do the addition the first column on the right sums to 62, carry the 6, that means you sum the next column plus 6, giving 57 carry the 5, next column is then 59, carry the 5, next column 67, carry the 6, next column, 76 carry the 7, next column, 56 there's no column to carry the 5 onto, and in the next step, it will be merged with the 2 (6 digits before), and then tossed because 7 creates a term that is 0 mod 7. Doing the same to other 7's and the nine gives 660200 we then replace x=10 with 3, via polynomial remainder theorem, and evaluate the sum shown above.Formula used $$sum_{n=0}^Ld_na^nequivsum_{n=0}^L(d_nbmod p)(a_nbmod p)^{(n bmod (p-1))} pmod p$$ we did the exponent part first, the base part second, and the coefficient (digit) part third, we then used the simple reduction mod p last.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    and had I not made an stupid error, with this small modulus, no calculator required.
    $endgroup$
    – Roddy MacPhee
    16 hours ago










  • $begingroup$
    I think this requires a bit more step-by-step explanation...
    $endgroup$
    – jcaron
    15 hours ago










  • $begingroup$
    All the steps are literally in there. but I guess I can show the summation instead of talking about it.
    $endgroup$
    – Roddy MacPhee
    15 hours ago










  • $begingroup$
    It also has a remainder of 2 if evaluated as hexadecimal.
    $endgroup$
    – Roddy MacPhee
    10 hours ago










  • $begingroup$
    Ouch! Your added explanation has confused me more than the original :-( . Maybe you could identify exactly what "first column on the right" is, and what exactly is summing to 62 (sum of the modulos?) . And so on. Or are you calculating , e.g., 115792 mod 7 and adding that to the sum of 10^k for that group?
    $endgroup$
    – Carl Witthoft
    9 hours ago














5












5








5





$begingroup$

$$begin{array}{cccccc}115792&089237&316195&423570&985008&687907\852837&564279&074904&382605&163141&518161\494336end{array}$$
Sum up the places of these numbers, by place value carrying when needed, then apply $10^kequiv 3^k bmod 7$ you'll then have a much smaller number to find the remainder of that's equivalent.
5667972, which goes to :$$6(3^5)+6(3^4)+2(3^2)equiv 1458+486+18equiv 2+3+4equiv 2 bmod 7$$ so the largest multiple of 7, is 2 less than the number. Yes, this is a slightly tedious way to go, but it's inspired via extension of Fermat's little theorem, and polynomial remainder theorem.



The reason I broke it into 6 digits at a time, is because Fermat's extension, is that exponents that have the same remainder mod p-1, will give back the same remainder with the same base. That means you can simply turn one into the other, adding like terms. you then go and do the addition the first column on the right sums to 62, carry the 6, that means you sum the next column plus 6, giving 57 carry the 5, next column is then 59, carry the 5, next column 67, carry the 6, next column, 76 carry the 7, next column, 56 there's no column to carry the 5 onto, and in the next step, it will be merged with the 2 (6 digits before), and then tossed because 7 creates a term that is 0 mod 7. Doing the same to other 7's and the nine gives 660200 we then replace x=10 with 3, via polynomial remainder theorem, and evaluate the sum shown above.Formula used $$sum_{n=0}^Ld_na^nequivsum_{n=0}^L(d_nbmod p)(a_nbmod p)^{(n bmod (p-1))} pmod p$$ we did the exponent part first, the base part second, and the coefficient (digit) part third, we then used the simple reduction mod p last.






share|cite|improve this answer











$endgroup$



$$begin{array}{cccccc}115792&089237&316195&423570&985008&687907\852837&564279&074904&382605&163141&518161\494336end{array}$$
Sum up the places of these numbers, by place value carrying when needed, then apply $10^kequiv 3^k bmod 7$ you'll then have a much smaller number to find the remainder of that's equivalent.
5667972, which goes to :$$6(3^5)+6(3^4)+2(3^2)equiv 1458+486+18equiv 2+3+4equiv 2 bmod 7$$ so the largest multiple of 7, is 2 less than the number. Yes, this is a slightly tedious way to go, but it's inspired via extension of Fermat's little theorem, and polynomial remainder theorem.



The reason I broke it into 6 digits at a time, is because Fermat's extension, is that exponents that have the same remainder mod p-1, will give back the same remainder with the same base. That means you can simply turn one into the other, adding like terms. you then go and do the addition the first column on the right sums to 62, carry the 6, that means you sum the next column plus 6, giving 57 carry the 5, next column is then 59, carry the 5, next column 67, carry the 6, next column, 76 carry the 7, next column, 56 there's no column to carry the 5 onto, and in the next step, it will be merged with the 2 (6 digits before), and then tossed because 7 creates a term that is 0 mod 7. Doing the same to other 7's and the nine gives 660200 we then replace x=10 with 3, via polynomial remainder theorem, and evaluate the sum shown above.Formula used $$sum_{n=0}^Ld_na^nequivsum_{n=0}^L(d_nbmod p)(a_nbmod p)^{(n bmod (p-1))} pmod p$$ we did the exponent part first, the base part second, and the coefficient (digit) part third, we then used the simple reduction mod p last.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 7 hours ago

























answered 17 hours ago









Roddy MacPheeRoddy MacPhee

418117




418117












  • $begingroup$
    and had I not made an stupid error, with this small modulus, no calculator required.
    $endgroup$
    – Roddy MacPhee
    16 hours ago










  • $begingroup$
    I think this requires a bit more step-by-step explanation...
    $endgroup$
    – jcaron
    15 hours ago










  • $begingroup$
    All the steps are literally in there. but I guess I can show the summation instead of talking about it.
    $endgroup$
    – Roddy MacPhee
    15 hours ago










  • $begingroup$
    It also has a remainder of 2 if evaluated as hexadecimal.
    $endgroup$
    – Roddy MacPhee
    10 hours ago










  • $begingroup$
    Ouch! Your added explanation has confused me more than the original :-( . Maybe you could identify exactly what "first column on the right" is, and what exactly is summing to 62 (sum of the modulos?) . And so on. Or are you calculating , e.g., 115792 mod 7 and adding that to the sum of 10^k for that group?
    $endgroup$
    – Carl Witthoft
    9 hours ago


















  • $begingroup$
    and had I not made an stupid error, with this small modulus, no calculator required.
    $endgroup$
    – Roddy MacPhee
    16 hours ago










  • $begingroup$
    I think this requires a bit more step-by-step explanation...
    $endgroup$
    – jcaron
    15 hours ago










  • $begingroup$
    All the steps are literally in there. but I guess I can show the summation instead of talking about it.
    $endgroup$
    – Roddy MacPhee
    15 hours ago










  • $begingroup$
    It also has a remainder of 2 if evaluated as hexadecimal.
    $endgroup$
    – Roddy MacPhee
    10 hours ago










  • $begingroup$
    Ouch! Your added explanation has confused me more than the original :-( . Maybe you could identify exactly what "first column on the right" is, and what exactly is summing to 62 (sum of the modulos?) . And so on. Or are you calculating , e.g., 115792 mod 7 and adding that to the sum of 10^k for that group?
    $endgroup$
    – Carl Witthoft
    9 hours ago
















$begingroup$
and had I not made an stupid error, with this small modulus, no calculator required.
$endgroup$
– Roddy MacPhee
16 hours ago




$begingroup$
and had I not made an stupid error, with this small modulus, no calculator required.
$endgroup$
– Roddy MacPhee
16 hours ago












$begingroup$
I think this requires a bit more step-by-step explanation...
$endgroup$
– jcaron
15 hours ago




$begingroup$
I think this requires a bit more step-by-step explanation...
$endgroup$
– jcaron
15 hours ago












$begingroup$
All the steps are literally in there. but I guess I can show the summation instead of talking about it.
$endgroup$
– Roddy MacPhee
15 hours ago




$begingroup$
All the steps are literally in there. but I guess I can show the summation instead of talking about it.
$endgroup$
– Roddy MacPhee
15 hours ago












$begingroup$
It also has a remainder of 2 if evaluated as hexadecimal.
$endgroup$
– Roddy MacPhee
10 hours ago




$begingroup$
It also has a remainder of 2 if evaluated as hexadecimal.
$endgroup$
– Roddy MacPhee
10 hours ago












$begingroup$
Ouch! Your added explanation has confused me more than the original :-( . Maybe you could identify exactly what "first column on the right" is, and what exactly is summing to 62 (sum of the modulos?) . And so on. Or are you calculating , e.g., 115792 mod 7 and adding that to the sum of 10^k for that group?
$endgroup$
– Carl Witthoft
9 hours ago




$begingroup$
Ouch! Your added explanation has confused me more than the original :-( . Maybe you could identify exactly what "first column on the right" is, and what exactly is summing to 62 (sum of the modulos?) . And so on. Or are you calculating , e.g., 115792 mod 7 and adding that to the sum of 10^k for that group?
$endgroup$
– Carl Witthoft
9 hours ago











1












$begingroup$

Just divide the number by 7, if the mod is 0, you subtract 1 from the quotient and multiply it by 7, else, the quotient times 7 is your desired number.



Ex:
70 / 7 = 10, with mod 0. 10-1 = 9 => 9 * 7 = 63 >Biggest multiple under 70.



71 / 7 = 10, with mod 1. 10 * 7 = 70 => Biggest multiple under 71






share|cite|improve this answer










New contributor




Fabio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$













  • $begingroup$
    That's only fast to a point, and only if you know your multiples.
    $endgroup$
    – Roddy MacPhee
    14 hours ago
















1












$begingroup$

Just divide the number by 7, if the mod is 0, you subtract 1 from the quotient and multiply it by 7, else, the quotient times 7 is your desired number.



Ex:
70 / 7 = 10, with mod 0. 10-1 = 9 => 9 * 7 = 63 >Biggest multiple under 70.



71 / 7 = 10, with mod 1. 10 * 7 = 70 => Biggest multiple under 71






share|cite|improve this answer










New contributor




Fabio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$













  • $begingroup$
    That's only fast to a point, and only if you know your multiples.
    $endgroup$
    – Roddy MacPhee
    14 hours ago














1












1








1





$begingroup$

Just divide the number by 7, if the mod is 0, you subtract 1 from the quotient and multiply it by 7, else, the quotient times 7 is your desired number.



Ex:
70 / 7 = 10, with mod 0. 10-1 = 9 => 9 * 7 = 63 >Biggest multiple under 70.



71 / 7 = 10, with mod 1. 10 * 7 = 70 => Biggest multiple under 71






share|cite|improve this answer










New contributor




Fabio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$



Just divide the number by 7, if the mod is 0, you subtract 1 from the quotient and multiply it by 7, else, the quotient times 7 is your desired number.



Ex:
70 / 7 = 10, with mod 0. 10-1 = 9 => 9 * 7 = 63 >Biggest multiple under 70.



71 / 7 = 10, with mod 1. 10 * 7 = 70 => Biggest multiple under 71







share|cite|improve this answer










New contributor




Fabio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this answer



share|cite|improve this answer








edited 14 hours ago





















New contributor




Fabio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









answered 14 hours ago









FabioFabio

112




112




New contributor




Fabio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Fabio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Fabio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • $begingroup$
    That's only fast to a point, and only if you know your multiples.
    $endgroup$
    – Roddy MacPhee
    14 hours ago


















  • $begingroup$
    That's only fast to a point, and only if you know your multiples.
    $endgroup$
    – Roddy MacPhee
    14 hours ago
















$begingroup$
That's only fast to a point, and only if you know your multiples.
$endgroup$
– Roddy MacPhee
14 hours ago




$begingroup$
That's only fast to a point, and only if you know your multiples.
$endgroup$
– Roddy MacPhee
14 hours ago










kpopguy is a new contributor. Be nice, and check out our Code of Conduct.










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kpopguy is a new contributor. Be nice, and check out our Code of Conduct.












kpopguy is a new contributor. Be nice, and check out our Code of Conduct.
















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