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Q.
The number of divisors of $ 3\times {{7}^{3}},7\times {{11}^{2}} $ and $ 2\times 61 $ are in:
Bihar CECEBihar CECE 2006
Solution:
Key Idea : If $a, b$ and $c$ are the prime numbers,
then the divisors of $a^{n_{1}} b^{n_{2}} c^{n_{3}}$ is
$\left(n_{1}+1\right)\left(n_{2}+1\right)\left(n_{3}+1\right)$
The number of divisors of $3 \times 7^{3}=8$,
the number of divisors of $7 \times 11^{2}=6$
and the number of divisors of $2 \times 61=4$
$\Rightarrow 8,6,4$ are in AP with common difference $-2$.