Numbers and Divisibility

Number Theory Level 2

Suppose a positive integer N is divisible by 21 and by 9. Find the total number of smallest positive integers that can divide N.

DETAILS AND ASSUMPTIONS:

1) Total number means, for example, the number 10 has the following divisors i.e, 1, 2, 5, 10. therefore it has a totally 4 divisors.

1 solution

A Former Brilliant Member
Feb 24, 2015

$N$ is a number that is divisible by both $21$ and $9$ $\rightarrow N = k\cdot LCM(21,9) ; k \in \quad \text{Natural Numbers }$ .

So ,the smallest such number is $63$ . WLOG let's take the number $N=63$ .

Now , $63 = 3^{2} \times 7$ , so the total number of it's factors $= (2+1)\cdot (1+1) = 6$