Question

Which of the following numbers will have an even number of factors?

• A. 1680
• B. 52900
• C. 30000
• D. 36100

Answer 1

Answer:

C. 30000

Concept used:

We know when a number is a perfect square if it has an odd number of factors.

For example, 25 is a perfect square then it has an odd number of factors that is 1,5,25.

No of factors = 3 (odd value)

Calculation:

As we know,

1600, 52900 and 36100 are perfect square.

So, these have an odd number of factors.

30000 is not a perfect square so it has an even number of factors.

⇒ 1600 = 5² $\times$ 2⁶ 

Total no of factors = (2 + 1) $\times$ (6 + 1) = 21 (odd value)

⇒ 52900 = 23² $\times$ 2² $\times$ 5² )

Total no of factors = (2 + 1) $\times$ (2 + 1) $\times$ (2 + 1) = 27 (odd value)

⇒ 30000 = 5⁴ $\times$ 2⁴ $\times$ 3 

Total no of factors = (4 + 1) $\times$ (4 + 1) $\times$ (1 + 1) = 50 (even value)

⇒ 36100 = 19² $\times$ 2² $\times$ 5² 

Total no of factors = (2 + 1) $\times$ (2 + 1) $\times$ (2 + 1) = 27 (odd value)

∴ The correct answer is 30000.