Find the number of prime factors in the product of 2512

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Find the number of prime factors in the product of 25¹² × 10⁷ × 14⁷.

1. 50
2. 52
3. 51
4. 54

Correct Option: B

We break each base number into prime factors.
Now, 25¹²
= (5 × 5)¹² = 5¹² × 5¹²
10⁷ = (2 × 5)⁷ = 2⁷ × 5⁷
and , 14⁷ = (2 × 7)⁷ = 2⁷ × 7⁷
∴ 25¹² × 10⁷ × 14⁷ = 5¹² × 5¹² × 2⁷ × 5⁷ × 2⁷ × 7⁷
= 5^{12 + 12 + 7} × 2⁷ + 7 × 7⁷ = 5³¹ × 2¹⁴ × 7⁷
∴ Number of prime factors = 31 + 14 + 7 = 52