Question

If \( n \) is the product of the integers from 1 to 8, inclusive, how many different prime factors greater than 1 does \( n \) have?

• A. Four
• B. Five
• C. Six
• D. Seven
• E. Eight

Hint:

To find the prime factors of a number, factorize the number into its prime components.

Answer 1

The Correct Option is C

Step 1: Find the prime factorization of \( n \).
We are asked to find the prime factors of \( n \), which is the product of integers from 1 to 8: \[ n = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 = 2^7 \times 3^2 \times 5 \times 7 \] Step 2: Count the distinct prime factors.
The prime factors of \( n \) are \( 2, 3, 5, 7 \), so there are 6 distinct prime factors greater than 1. \[ \boxed{6} \]