Question

In how many ways can the number \(7056\) be resolved into two factors?

• A. \(20\)
• B. \(21\)
• C. \(23\)
• D. \(22\)

Hint:

If a number has (n) factors:

Factor pairs (= fracn2) for non-square numbers
Factor pairs (= fracn+12) for perfect squares

Answer 1

The Correct Option is C

Step 1: Express \(7056\) as a perfect square. \[ 7056 = 84^2 \] Step 2: Find the prime factorisation of \(7056\). \[ 84 = 2^2 \times 3 \times 7 \] \[ 7056 = (2^2 \times 3 \times 7)^2 = 2^4 \times 3^2 \times 7^2 \] Step 3: Find the total number of factors. If \(N = p^a q^b r^c\), then number of factors: \[ = (a+1)(b+1)(c+1) \] \[ = (4+1)(2+1)(2+1) = 5 \times 3 \times 3 = 45 \] Step 4: Find the number of ways to resolve into two factors. Since \(7056\) is a perfect square, the number of distinct factor pairs is: \[ \frac{45 + 1}{2} = 23 \] Step 5: Final conclusion. The number \(7056\) can be resolved into two factors in \(\boxed{23}\) ways.