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Number Theory Level 3

We know the following:

A positive integer that ends in 1 is always divisible by 1.
A positive integer that ends in 2 is always divisible by 2.
A positive integer that ends in 5 is always divisible by 5.

Similarly, how many integers $n\, (10 \leq n < 100)$ are there such that any positive integer ending in $n$ is always divisible by $n?$

Bonus: In general, for a positive integer $p,$ how many integers $n\, \big(10^p \leq n < 10^{p+1}\big)$ are there such that any positive integer ending in $n$ is always divisible by $n?$

4 5 6 7

1 solution

Stephen Brown
Dec 28, 2017

All $n$ being considered are two digits, so another way to frame the problem is to find all n such that, for all integers $k$ :

$n+100k \equiv 0\bmod{n} \Rightarrow 100k \equiv 0 \bmod{n} \Rightarrow n|100$

Thus the only possible $n$ are $10, 20, 25, 50$ , a total of $\boxed{4}$ options.

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