Question

How many factors of the number 2⁴ × 5³ × 15³ × 10⁴ × 4³, that are multiples of 225?

1. 660
2. 225
3. 135
4. 270

Answer 1

Correct Answer - Option 4 : 270

Concept Used:

I if x = kY where x, Y and k are integers.

Number of multiples of k = Number of factors of Y

II N = a^p × b^q × c^r × …

Number of factors of N = (p + 1) × (q + 1) × (r + 1) × … Where a, b and c are prime factors of N.

Calculation:

225 = 3² × 5²

Given expression

2⁴ × 5³ × 15³ × 10⁴ × 4³ = 2⁴ × 5³ × 3³ × 5³ × 5⁴ × 2⁴ × 2⁶

⇒ 2¹⁴ × 3³ × 5¹⁰

⇒ 3² × 5²(2¹⁴ × 3 × 5⁸)

Number of factors of 2¹⁴ × 3 × 5⁸ = (14 + 1) × (1 + 1) × (8 + 1)

⇒ 15 × 2 × 9

⇒ 270

∴ Number of multiples of 225 = Number of factors of 2¹⁴ × 3 × 5⁸ = 270