Question

# How many factors of ${2^5} \times {3^6} \times {5^2}$ are perfect squares?A. 24B. 12C. 16D. 22

Hint: First of all, find the possible number of ways in which perfect square factors of ${2^5},{3^6},{5^2}$ can be arranged individually. Then use the multiplicative principle of permutations to get the required answer. So, use this concept to reach the solution of the given problem.

For a perfect square, the power of each should be even.
The possible factors of ${2^5}$ are ${2^0},{2^1},{2^2},{2^3},{2^4},{2^5}$
So, the possible perfect square factors of ${2^5}$ are ${2^0},{2^2},{2^4}$.
Therefore, possible number of ways of arranging the perfect square factors of ${2^5}$ = 3
The possible factors of ${3^6}$ are ${3^0},{3^1},{3^2},{3^3},{3^4},{3^5},{3^6}$
So, the possible perfect square factors of ${3^6}$ are ${3^0},{3^2},{3^4},{3^6}$.
Therefore, possible number of ways of arranging the perfect square factors of ${3^6}$ = 4
The possible factors of ${5^2}$ are ${5^0},{5^1},{5^2}$
So, the possible perfect square factors of ${5^2}$ are ${5^0},{5^2}$.
Therefore, possible number of ways of arranging the perfect square factors of ${5^2}$ = 2
By using multiplicative principle of permutations, we have
The total number of ways of arranging the perfect square factors of ${2^5} \times {3^6} \times {5^2}$ are $3 \times 4 \times 2 = 24$
Hence there are 24 factors of ${2^5} \times {3^6} \times {5^2}$ which are perfect squares.
Thus, the correct option is A. 24

Note: In this problem we have used multiplicative principle permutations i.e., if there are $x$ number of ways of arranging one thing and$y$ number of ways of arranging another, then the total number of ways of arranging both the things is given in $xy$ number of ways.