Question

Two positive integers a and b can be written as a = x³y² and b = xy³, where x and y are prime numbers. Find HCF(a, b) and LCM(a, b).

Answer 1

1. Analyse given factorisations

The integers are expressed using prime bases x and y:

a = x³y²

b = x¹y³

2. Determine the HCF

To find the Highest Common Factor, take the lowest power of each common prime factor:

For x: The lower exponent between 3 and 1 is 1 (→ x¹).

For y: The lower exponent between 2 and 3 is 2 (→ y²).

HCF(a, b) = x¹ × y²

= xy²

3. Determine the LCM

To find the Least Common Multiple, take the highest power of each prime factor:

For x: The highest exponent between 3 and 1 is 3 (→ x³).

For y: The highest exponent between 2 and 3 is 3 (→ y³).

LCM(a, b) = x³ × y³

= x³y³