A fifty digit positive integer satisfies all these given conditions:
- The number is constituted entirely by the digit 1 with the exception of the units digit.
- The number is divisible by 53.
Determine the units digit.
Extra Challenge: Solve this puzzle without using a computer-assisted method.
From Fermat's Little Theorem we have:
10^52 = 1 mod 53
Subtract 1 from each side, multiply both sides by 6, and factor the left side:
54*(111...111) = 0 mod 53
[52 1's]
54=1 mod 53, so that factor drops out.
Subtract 11 from each side then multiply both sides by 44, and then factor:
4400*(111...111) = -484 mod 53
[50 1's]
4400 = 1 mod 53 and -484 = -7 mod 53. Then:
111...111 = -7 mod 53
[50 1's]
Now just add 7 to each side to make:
111..118 = 0 mod 53
[49 1's]
The left side is now a number that satisfies all the conditions. Its units digit is 8.
Solution
