Question

If $P=2^4\cdot 6^3\cdot 5^2\cdot {15}^2$, then the number of proper even divisors of $P$ will be

• A. $119$
• B. $120$
• C. $209$
• D. $210$

Answer 1

The correct option is C $209$

$P=2^4\cdot 6^3\cdot 5^2\cdot {15}^2$
$\Rightarrow P=2^4\cdot (2\times 3{)}^3\cdot 5^2\cdot (3\times 5{)}^2\Rightarrow P=2^7\cdot 3^5\cdot 5^4$
Total divisors
$=(7+1)(5+1)(4+1)=240$
Total odd divisors
$=(5+1)(4+1)=30$
Total even divisors
$=240-30=210$
Total proper even divisors
$=210-1=209$
$(\because$ the number itself will not be included$)$