Discussion
If 3024 = p4 × q3 × r, where p, q and r are prime numbers, then what is the value of (3p − 5q + r) ?
Answer: B
The value is −2.
Factorise 3024 into primes:
\[ \begin{aligned} 3024 &= 2 \times 1512 = 2^{4} \times 189 \\ 189 &= 3^{3} \times 7 \\ 3024 &= 2^{4} \times 3^{3} \times 7 \end{aligned} \]
Comparing with \(p^{4} \times q^{3} \times r\) gives p = 2, q = 3 and r = 7.
\[ 3p - 5q + r = 6 - 15 + 7 = -2 \]
Swapping p and q by mistake (p = 3, q = 2) gives 9 − 10 + 7 = 6, which is not an option, so the factorisation must be read with the fourth power on 2.
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