Discussion
If 313632 = \(p^{2} \times q^{5} \times r^{4}\), where p, q and r are prime numbers, then what is the value of (p + q - 2r) ?
Answer: B
The value of (p + q - 2r) is 7.
Factorise 313632 into primes:
\[ \begin{aligned} 313632 &= 2^{5} \times 9801 \\ 9801 &= 99^{2} = 3^{4} \times 11^{2} \\ 313632 &= 11^{2} \times 2^{5} \times 3^{4} \end{aligned} \]
Matching the powers, p = 11, q = 2 and r = 3, so p + q - 2r = 11 + 2 - 6 = 7.
The key step is matching each prime to its exponent: 2 is the prime raised to the fifth power, 3 to the fourth and 11 to the second. Swapping any two of them changes the result, so 8, 9 and 6 do not fit.
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