Question

The number of divisors of \(7!\) is:

• A. \(72\)
• B. \(24\)
• C. \(64\)
• D. \(60\)

Hint:

To find the number of divisors of \(n!\), first perform prime factorization of the factorial, then apply the divisor count formula: multiply one more than each of the exponents.

Answer 1

The Correct Option is D

Step 1: Compute the value of \(7!\):
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \] Step 2: Prime factorisation of 5040:
\[ 5040 = 2^4 \times 3^2 \times 5 \times 7 \] Step 3: Use the formula for total number of divisors of a number:
If \(n = p_1^{a_1} \cdot p_2^{a_2} \cdot \ldots \cdot p_k^{a_k}\), then the number of divisors of \(n\) is:
\[ (a_1 + 1)(a_2 + 1)\ldots(a_k + 1) \] Step 4: Apply the formula:
\[ \text{Number of divisors} = (4+1)(2+1)(1+1)(1+1) = 5 \times 3 \times 2 \times 2 = 60 \] \[ \boxed{60} \text{ is the number of divisors of } 7! \]