Question

How many factors of $2^4\times3^5\times10^4$ are perfect squares which are greater than 1 ?

• A. 44
• B. 38
• C. 45
• D. 22

Answer 1

Answer: (A)

44

Answer 2

$2^4×3^5×10^4=2^8×3^5×5^4$
To obtain perfect squares, we should consider solely the even powers of the prime factors within the number.
There are five possible ways to utilize the prime factor 2i.e, $2^0, 2^2, 2^4, 2^6, 2^8$
There are three possible ways to utilize the prime factor 3 i.e. $3^0, 3^2, 3^4$
There are three possible ways to utilize the prime factor 5 i.e. $5^0, 5^2, 5^4$
Hence, the overall count of factors that qualify as perfect squares $= 5×3×3=45$
However, this count encompasses the number 1. Therefore, when excluding 1, the desired quantity is $45 - 1 = 44.$