Question

Find the LCM and HCF of the following integers by applying the prime factorisation method:

26, 65 and 117

Answer 1

Given: 26, 65 and 117

Step-wise calculation:

1. Prime factorisation (by the prime-factorisation method).

26 = 2 × 13

= 2¹ × 13¹

65 = 5 × 13

= 5¹ × 13¹

117 = 3 × 39

= 3 × 3 × 13

= 3² × 13¹

2. HCF (highest common factor), take each common prime with the minimum exponent:

Common prime is 13 with exponent min(1, 1, 1) = 1

HCF = 13

As a check, the known HCF of 65 and 117 is 13, consistent with the above.

3. LCM (least common multiple), take each prime appearing in any factorisation with the maximum exponent:

Primes and max exponents: 2¹, 3², 5¹, 13¹

LCM = 2¹ × 3² × 5¹ × 13¹

= 2 × 9 × 5 × 13

= 1170

Quick divisibility check:

1170 ÷ 26 = 45,

1170 ÷ 65 = 18,

1170 ÷ 117 = 10 (all integers), so 1170 is a common multiple.