Question

The number of even divisors of the number $N = 12600 = {2^3}{3^2}{5^2}7$ is
a. 72
b. 54
c. 18
d. None of these

Answer 1

Here we are going to find prime factors and segregating all the factors into the even power and the odd power. Prime numbers are the whole number greater than 1 which cannot be made by multiplying other whole numbers. E.g. $2,3,5,7,....$

Complete step by step solution:
On prime factorization, it consist of - $12600 = {2^3} \times {3^2} \times {5^2} \times 7$

According to general concept total number of factors in case of ${x^n}$= (n+1)
Number of factors = $(3 + 1)(2 + 1)(2 + 1)(1 + 1)$
$= 72$
Removing power of 2 keeping remaining factors which are odd is -
$\begin{array}{l} = (2 + 1)(2 + 1)(1 + 1)\\\ = (3)(3)(2)\\\ = 18 \end{array}$
As, total number of factors= the number of even factors + the number of odd factors
To get even factors we would subtract odd factors from the total number of factors.
Therefore, Even factors
$\begin{array}{l} = 72 - 18\\\ = 54 \end{array}$
**Hence, the required answer is 54.

Therefore, option (B), 54 is correct.**

Note: Prime factorization is finding which prime numbers multiply together to make the original number. Even numbers are $2,4,6,....$ whereas odd numbers are $1,3,5,....$ Remember 1 is neither prime nor composite. Always remember the difference between prime numbers $(2,3,5....)$ and composite numbers $(4,6,8,...)$ to solve factorization examples.