IPMAT Indore 2025 (SA) - The number of factors of 3^5 x 5^8 x 7^2 that are perfect squares is | PYQs + Solutions

IPMAT Indore 2025

Number System

Factorisation

Easy

The number of factors of $3^5 \times 5^8 \times 7^2$ that are perfect squares is

Entered answer:

✅ Correct Answer: 30

A perfect square factor will have the form

3^a \times 5^b \times 7^c

where each exponent must be even.

This is because a perfect square is a number that can be expressed as

n^2

for some integer

n

For our number

3^5 \times 5^8 \times 7^2

, we need to find all possible even exponents that don't exceed the original exponents:

- For

3^5

: The exponent

a

can be 0, 2, 4 (3 possibilities)

\newline

- For

5^8

: The exponent

b

can be 0, 2, 4, 6, 8 (5 possibilities)

\newline

- For

7^2

: The exponent

c

can be 0, 2 (2 possibilities)

Using the multiplication principle, the total number of perfect square factors is:

3 \times 5 \times 2 = 30

Therefore, the number of factors of

3^5 \times 5^8 \times 7^2

that are perfect squares is 30.