Compute the total number of prime factors, counting multiplicity, of the product 8^20 × 15^24 × 7^15.

Introduction / Context:
“Total number of prime factors” here means counting prime factors with multiplicity (the sum of exponents in the prime factorization). We factor each base, multiply exponents appropriately, and then add the exponents.

Given Data / Assumptions:

• Expression: 8^20 × 15^24 × 7^15
• We count prime factors with multiplicity

Concept / Approach:
Prime-factorize each base: 8 = 2^3, 15 = 3*5, and 7 is prime. Then raise to the given powers and add exponents across all primes involved.

Step-by-Step Solution:
8^20 = (2^3)^20 = 2^60 ⇒ contributes 60 twos.15^24 = (3*5)^24 = 3^24 * 5^24 ⇒ contributes 24 threes and 24 fives.7^15 ⇒ contributes 15 sevens.Total prime factors with multiplicity = 60 + 24 + 24 + 15 = 123.

Verification / Alternative check:
No overlap among these primes beyond what is counted in exponents. Summation is straightforward.

Why Other Options Are Wrong:
59 and 98 underestimate by dropping exponents; 138 overestimates; 117 misses some contribution.

Common Pitfalls:
Confusing “total prime factors” with “number of distinct primes” (which would be 4 here: 2, 3, 5, 7).

Final Answer:
123