Consecutive divisibility

Number Theory Level 2

$n$ is divisible by 4.
$n+1$ is divisible by 4.
$n+2$ is divisible by 4.
$n+3$ is divisible by 4.
$n+4$ is divisible by 4.

If precisely one of the statements above is correct, then which of the following statements must be wrong?

n+5

is divisible by 4.

n+6

is divisible by 4.

n+7

is divisible by 4.

n+8

is divisible by 4.

1 solution

Zach Abueg
Jul 5, 2017

If $4 \mid n$ , then $4 \mid n + 4$ . In general, if $4 \mid n$ then $4 \mid n + 4m \ \forall \ m \in \mathbb{N}$ . However, only one of the statements is correct. Since they cannot both be correct, they must be wrong.

Thus, $\boxed{4 \not \mid n + 8}$ .

Consecutive divisibility

1 solution

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