Find the number of prime factors in the product (30)5 × (24)5.1. 352.

1.	Find the number of prime factors in the product (30)5 × (24)5.1. 352. 453. 104. 30
Answer» Correct Answer - Option 1 : 35 Concept used: a^x × a^y = a^{x + y} If a, b and c are prime numbers, then the number of prime factors of ax × by × c^z is (x + y + z). Calculation: Prime factors of 30 = 2 × 3 × 5 Prime factors of 24 = 2 × 2 × 2 × 3 (30)5 × (24)5 ⇒ (2 × 3 × 5)⁵ × (2 × 2 × 2 × 3)⁵ ⇒ 2⁵ × 3⁵ × 5⁵ × 2⁵ × 2⁵ × 2⁵ × 3⁵ ⇒ (2)^{5 + 5 + 5 + 5} × (3)^{5 + 5} × (5)⁵ ⇒ (2)20 × (3)10 × (5)5 Number of prime factors = 20 + 10 + 5 = 35 ∴ The number of prime factors in the product (30)5 × (24)5 are 35.

Find the number of prime factors in the product (30)5 × (24)5.1. 352. 453. 104. 30

Answer» Correct Answer - Option 1 : 35

Concept used:
a^x × a^y = a^{x + y}
If a, b and c are prime numbers, then the number of prime factors of ax × by × c^z is (x + y + z).

Calculation:
Prime factors of 30 = 2 × 3 × 5
Prime factors of 24 = 2 × 2 × 2 × 3
(30)5 × (24)5
⇒ (2 × 3 × 5)⁵ × (2 × 2 × 2 × 3)⁵
⇒ 2⁵ × 3⁵ × 5⁵ × 2⁵ × 2⁵ × 2⁵ × 3⁵
⇒ (2)^{5 + 5 + 5 + 5} × (3)^{5 + 5} × (5)⁵
⇒ (2)20 × (3)10 × (5)5
Number of prime factors = 20 + 10 + 5 = 35
∴ The number of prime factors in the product (30)5 × (24)5 are 35.

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