Question

2
Exercise - 3.7
1) Find the LCM and HCF of the following numbers and check their relationship
(i) 15, 24
(ii) 8, 25 (iii) 12, 48 (iv) 30, 48​

Answer 1

HCF: It stands for, 'Highest Common Factor'. It is defined as product of common prime factors of smallest power.

LCM: It stands for, 'Lowest Common Multiple'. It is defined as product of greatest power of prime factors involved in the numbers.

We can verify that for any two positive integers a and b, HCF(a,b)×LCM(a,b) = a×b.

Let us solve each question.

(i) 15,24

→ Prime factors of 15 are = 3×5

→ Prime factors of 24 are = 2³×3

• HCF(15,24) = 3

• LCM(15,24) = 2³×3×5 = 8×3×5 = 120

Verification:

→ HCF(15,24)×LCM(15,24) = 15×24

→ 3×120 = 15×24

→ 360 = 360 _______[Verified]

(ii) 8,25

→ Prime factors of 8 are = 2³

→ Prime factors of 25 are = 5²

• HCF(8,25) = 1. There are no other common factors for 8 and 25 other than 1.

• LCM(8,25) = 2³×5² = 200

Verification:

→ HCF(8,25)×LCM(8,25) = 8×25

→ 1×200 = 8×25

→ 200 = 200 _______[Verified]

(iii) 12,48

→ Prime factors of 12 are = 2²×3

→ Prime factors of 48 are = $2^4$×3

• HCF(12,48) = 12

• LCM(12,48) = $2^4$×3 = 48

Verification:

→ HCF(12,48)×LCM(12,48) = 12×48

→ 12×48 = 12×48 _______[Verified]

(iv) 30,48

→ Prime factors of 30 = 2×3×5

→ Prime factors of 48 = $2^4$×3

• HCF(30,48) = 2×3 = 6

• LCM(30,48) = $2^4$×3×5 = 240

Verification:

→ HCF(30,48)×LCM(30,48) = 30×48

→ 6×240 = 30×48

→ 1440 = 1440 _______[Verified]