Determine the minimum possible value of a base ten positive integer N, such that:

1. N is constituted entirely by 0s and 1s, and:
2. N is divisible by 2475.

Extra Challenge: Solve this puzzle without using a computer program.

2475 = 3^2 * 5^2 * 11

Merely two trailing zeros will provide for the 5^2. We need concern ourselves only with mod 99.

100 mod 99 = 1
1000 mod 99 = 10

and in fact the powers of ten mod 99 alternate thereafter.

Then:

1100 mod 99 = 11
110000 mod 99 = 11, etc.

We just need 9 pairs of 1's before two zeros to bring that to 99 mod 99 = 0 mod 99; that is, 18 1's followed by two zeros: 11111111111111111100.

 

Posted by Charlie on 2013-12-20 17:32:31