Question

1. the numbers 144 and 160 when written as the products of their prime factors, are 144 = 2⁴ × 3² and 160 = 2⁵ × 5. find the highest common factor and the lowest common multiple of 144 and 160.

Explanation:

Step1: Find HCF via shared primes

Take the lowest power of common prime factors.
Common prime is $2$, lowest power is $2^4$.
$\text{HCF} = 2^4 = 16$

Step2: Find LCM via all primes

Take the highest power of all primes present.
Primes are $2, 3, 5$; highest powers are $2^5, 3^2, 5^1$.
$\text{LCM} = 2^5 \times 3^2 \times 5 = 32 \times 9 \times 5$

Step3: Calculate LCM value

Multiply the terms step by step.
$32 \times 9 = 288$, $288 \times 5 = 1440$

Answer:

Highest Common Factor (HCF) = 16
Lowest Common Multiple (LCM) = 1440