Question

find the HCF and Lcm of 24 and 360 and verify HCF × LCm = products of two numbers​

Answer 1

Question :

find the HCF and Lcm of 24 and 360 and verify HCF × LCm = products of two numbers .

Answer :

• Two numbers are 24 and 360 .

• LCM and HCF ?

24 = 12 × 2

24 = 2 × 6 × 2

24 = 2 × 2 × 2 × 3

24 = $\sf {{2}^{3}\times 3}$

360 = 36 × 10

360 = 6 × 6 × 5 × 2

360 = 2 × 3 × 2 × 3 × 5 × 2

360 = $\sf {{2}^{3}\times {3}^{2}\times 5}$

HCF = Product of smallest power of common prime factors

HCF = $\sf {{2}^{3}\times 3}$

HCF = 8 × 3

HCF = 24

LCM = product of greatest power of each prime factor .

LCM = $\sf {{2}^{3}\times {3}^{2}\times 5}$

LCM = 8 × 9 × 5

LCM = 72 × 5

LCM = 360

Verification

LCM × HCF = product of two numbers

24 × 360 = 24 × 360

LHS = RHS

hence verified .

Answer 2

✿ HCF = 24

✿ LCM = 360

Step-by-step explanation:

Given:

• Two numbers: 24 and 360.

Things to do:

• HCF(24,360)
• LCM(24,360)
• We need to verify that, HCF(24,360)×LCM(24,360) = 24×360

Solution:

First, let us find the HCF of 24 and 360.

Prime factors of 24:

$\implies{\rm}$ 2×2×2×3

$\implies{\rm}$ 2³×3¹

Prime factors of 360:

$\implies{\rm}$ 2×2×2×3×3×5

$\implies{\rm}$ 2³×3²×5¹

HCF is the product of common prime factors which has the least power.

From the above method, we can see that 2 and 3 are the common prime factors.

Also, 2³ and 3¹ are the common prime factors with least power. Therefore:

$\implies{\rm}$ HCF(24,360) = 2³×3¹

$\implies{\rm}$ HCF(24,360) = 8×3

$\implies{\rm}$ HCF(24,360) = 24

LCM is the product of prime factors which has the highest power.

From the above method, 2³, 3² and 5¹ are the prime numbers with highest power. Therefore:

$\implies{\rm}$ LCM(24,360) = 2³×3²×5¹

$\implies{\rm}$ LCM(24,360) = 8×9×5

$\implies{\rm}$ LCM(24,360) = 360.

Hence, we have found the HCF and LCM of 24 and 360. Let us verify:

Verification:

We know that,

➙ HCF(a,b)×LCM(a,b) = a×b

➙ HCF(24,360)×LCM(24,360) = 24×360

➙ 24×360 = 24×360

➙ 8640 = 8640

Hence, verified. (LHS = RHS)