2006 Cyprus MO/Lyceum/Problem 24

Problem

The number of divisors of the number $2006$ is

$\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 4\qquad\mathrm{(C)}\ 8\qquad\mathrm{(D)}\ 5\qquad\mathrm{(E)}\ 6$

Solution

$2006=2\cdot17\cdot59$. A divisor of $2006$ is therefore in the form $2^m\cdot 17^n\cdot 59^p$, where $m\leq 1$, $n\leq 1$, and $p\leq 1$.

There are 2 choices for $m$, 2 choices for $n$, and 2 choices for $p$. Therefore, there are $2\cdot2\cdot2=\boxed{8}$ divisors of $2006$.

See also

2006 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 23
Followed by
Problem 25
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