Compute the H.C.F. of (22 × 3 × 5 × 74), (23 × 32 × 52 × 73) and (22 × 5 × 75).

Concept

The Highest Common Factor (H.C.F.) of numbers given in prime-factorised form is built from only those primes that appear in every number, each raised to its lowest exponent among the numbers. A prime missing from even one number cannot divide all of them, so it is excluded entirely.

Application

Write the three numbers and examine each prime base in turn:

• Number 1: 22 × 3 × 5 × 74

• Number 2: 23 × 32 × 52 × 73

• Number 3: 22 × 5 × 75

1. Prime 2 appears in all three with exponents 2, 3, 2; the lowest is 22.

2. Prime 3 appears only in Numbers 1 and 2 (it is absent from Number 3), so it is dropped.

3. Prime 5 appears in all three with exponents 1, 2, 1; the lowest is 51.

4. Prime 7 appears in all three with exponents 4, 3, 5; the lowest is 73.

Multiply the surviving prime powers: H.C.F. = 22 × 5 × 73 = 4 × 5 × 343 = 6860.

Cross-check

6860 = 22 × 5 × 73. Each prime power here does not exceed the corresponding power in any of the three numbers, so 6860 divides all three, and no larger common factor exists because raising any exponent would exceed it in at least one number.