Question

If $N={12}^3\times 3^4\times 5^2$, then the total number of even factors of N is

• A. $25$
• B. $121$
• C. $144$
• D. $84$

Answer 1

The correct option is C $144$

$N=(2^2\times 3{)}^3\times 3^4\times 5^2$
$=2^6\times 3^7\times 5^2$
$\therefore$ Total number of factors of N
$=(6+1)\times (7+1)\times (2+1)$
$=7\times 8\times 3=168$
Some of these are odd factors and some of these are even factors .The odd factors are formed with the combination of 3s and 5s
$\therefore$ Total number of odd factors
= ( 7 + 1 ) x (2 + 1) = 8 x 3 = 24
$\therefore$ Number of even factors = 168 - 24 = 144