If 313632 = p2 × q5 × r4, where p, q and r are prime numbers, then what is the…

Concept

Every integer has a unique prime factorisation: it can be written as a product of prime powers in exactly one way (Fundamental Theorem of Arithmetic). So to find unknown primes from a form like p2 × q5 × r4, factor the number completely, then match each prime to the base whose exponent equals the exponent shown for it.

Application

1. Factor 313632 by repeated division: 313632 = 25 × 34 × 112.

2. Match exponents to the given form p2 × q5 × r4. The base with exponent 2 is p, with exponent 5 is q, with exponent 4 is r.

3. Exponent 2 belongs to 11, so p = 11. Exponent 5 belongs to 2, so q = 2. Exponent 4 belongs to 3, so r = 3.

4. Substitute into p + q − 2r = 11 + 2 − 2 × 3 = 13 − 6 = 7.

Cross-check

Verify the factorisation: 112 × 25 × 34 = 121 × 32 × 81 = 313632, which matches the given number, so the prime assignments are correct and p + q − 2r = 7.