Dr. Divisible

Number Theory Level 2

Given that N N is a positive integer . If

  • N + 1 N+1 is divisible by 2,
  • N + 2 N+2 is divisible by 3,
  • N + 3 N+3 is divisible by 5, and
  • N + 4 N+4 is divisible by 7.

Is N + 18 N+18 divisible by 35?

Yes No

Skye Rzym by SKYE RZYM , 20, Indonesia

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1 solution

Brian Moehring
Mar 1, 2017

Note that N + 18 = ( N + 3 ) + 15 is divisible by 5 N + 18 = ( N + 4 ) + 14 is divisible by 7 \begin{aligned}N+18 &= (N+3) + 15 \text{ is divisible by } 5 \\ N+18 &= (N+4) + 14 \text{ is divisible by } 7\end{aligned}

Therefore, we can conclude N + 18 N+18 is divisible by lcm ( 5 , 7 ) = 35 \mbox{lcm}(5,7) = 35 .

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