Question

Find the H.C.F. and L.C.M. of 408 and 312.

Answer 1

Given: 408 and 312.

Step-wise calculation:

1. Prime factorise both numbers:

408 = 2 × 204

= 2² × 102 

= 2³ × 51

= 2³ × 3 × 17

312 = 2 × 156

= 2² × 78

= 2³ × 39

= 2³ × 3 × 13

2. H.C.F. (G.C.D.) = product of smallest powers of common primes:

Common primes: 2³ and 3¹

H.C.F. = 2³ × 3 

= 8 × 3 

= 24

3. L.C.M. = product of greatest powers of all primes appearing:

Primes involved: 2³, 3¹, 17¹, 13¹ 

L.C.M. = 2³ × 3 × 17 × 13

Compute: 2³ × 3 = 24

17 × 13 = 221

So, L.C.M. = 24 × 221 

= 5304

4. Quick check using HCF × LCM = product of the two numbers:

24 × 5304 = 127, 296 and 408 × 312 = 127, 296, so the values agree with the identity.

H.C.F. (408, 312) = 24

L.C.M. (408, 312) = 5304