GMAT Practice Question: How many positive integer divisors does 8 ! have?
Question
How many positive integer divisors does 8 ! have?
- 14
- 15
- 24
- 56
- 96
Topics: factorials, prime factorization, divisibility
Solution
Step 1
We prime factorize 8! by expressing each integer factor from 1 to 8 in its prime form
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
Step 2
We combine like prime factors by summing their exponents
8! = (2^3) \times 7 \times (2 \times 3) \times 5 \times (2^2) \times 3 \times 2 \times 1 = 2^{7} \times 3^{2} \times 5^{1} \times 7^{1}
Step 3
We count the number of positive divisors by adding one to each exponent and multiplying
(7+1) \times (2+1) \times (1+1) \times (1+1) = 8 \times 3 \times 2 \times 2 = 96
Answer
E
