How fast can you crack this code?

Level 2

My lock's combination is the smallest positive integer N N such that N N is a multiple of 3, N + 1 N+1 is a multiple of 5, N + 2 N+2 is a multiple of 7, N + 3 N+3 is a multiple of 9, and N + 4 N+4 is a multiple of 11.

What is N ? N?


The answer is 1734.

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1 solution

Gabriel Chacón
Apr 8, 2018

Although the problem can be solved using modular arithmetic, it is shorter to try and guess the solution. The condition ' N is a multiple of 3' is redundant, being that N +3 is a multiple of 9. So we start by finding the two smallest consecutive numbers that are multiples of 5 and 7 respectively: 20 and 21 . If we add 5·7=35 to these numbers, we obtain higher consecutive numbers that are multiples of 5 and 7 respectively. After four tries, we reach a third consecutive number that is a multiple of 9: 160, 161 and 162 . Now, by adding 5·7·9=315 a few times (5), we get a fourth consecutive number that is a multiple of 11: 1735,1736, 1737 and 1738 , which is 4 units greater than the combination we are looking for: 1734 .

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