Question

Let $N=3^5\times 5^4\times 6^9\times 7^7.$ The number of factors of $N$ which are perfect squares is

• A. $168$
• B. $180$
• C. $210$
• D. $480$

Answer 1

The correct option is C $210$

$N=3^5\times 5^4\times 6^9\times 7^7$
$=2^9\times 3^{14}\times 5^4\times 7^7$

A perfect square of $N$ is of the form $2^a\times 3^b\times 5^c\times 7^d$
where $a\in \{0,2,4,6,8\}$
$b\in \{0,2,4,\cdots ,14\}$
$c\in \{0,2,4\}$
$d\in \{0,2,4,6\}$

Total number is $5\times 8\times 3\times 4=480$