Question : Find the number of prime factors in the product (30)5 × (24)5.
Option 1: 35
Option 2: 30
Option 3: 45
Option 4: 10
Correct Answer: 35
Solution :
Given: The product is (30)
5
× (24)
5
.
If $a$, $b$, and $c$ are prime numbers, then the number of prime factors of $a^x\times b^y\times c^z$ is $(x + y + z)$ where $x,y$ and $z$ are positive integers.
(30)
5
× (24)
5
= (2 × 3 × 5)
5
× (2 × 2 × 2 × 3)
5
⇒ (30)
5
× (24)
5
= 2
5
× 3
5
× 5
5
× 2
5
× 2
5
× 2
5
× 3
5
⇒ (30)
5
× (24)
5
= 2
5+5+5+5
× 3
5+5
× 5
5
⇒ (30)
5
× (24)
5
= 2
20
× 3
10
× 5
5
The number of prime factors = 20 + 10 + 5 = 35.
Hence, the correct answer is 35.
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