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Q.
The number of positive divisors of $252$ is
ManipalManipal 2008
Solution:
We know that, if $a=p_{1}^{\alpha_{1}} \cdot p_{2}^{\alpha_{2}}$
Then, the total number of positive divisors of $a$ is
$T(a)=\left(\alpha_{1}+1\right)\left(\alpha_{2}+1\right) \ldots$
Given, $252=2^{2} \times 3^{2} \times 7^{1}$
Here, $\alpha_{1}=2, \alpha_{2}=2, \alpha_{3}=1$
$\therefore T(a)=(2+1)(2+1)(2+1)$
$=3 \cdot 3 \cdot 2$
$=18$