Question

Using prime factorization, find the HCF and LCM of 

(i) 36, 84 

(ii) 23, 31 

(iii) 96, 404 

(iv) 144, 198 

(v) 396, 1080 

(vi) 1152, 1664 

In each case verify that HCF × LCM = product of given numbers.

Answer 1

(i) Prime factorization: 

36 = 2² × 3 

84 = 2² × 3 × 7 

HCF = product of smallest power of each common prime factor in the numbers 

= 2² × 3 = 12 

LCM = product of greatest power of each prime factor involved in the numbers 

= 2² × 3² × 7 = 252 

(ii) Prime factorization: 

23 = 23 

31 = 31 

HCF = product of smallest power of each common prime factor in the numbers = 1 

LCM = product of greatest power of each prime factor involved in the numbers 

= 23 × 31 = 713 

(iii) Prime factorization: 

96 = 2⁵ × 3 

404 = 2² × 101 

HCF = product of smallest power of each common prime factor in the numbers 

= 2² = 4 

LCM = product of greatest power of each prime factor involved in the numbers 

= 2⁵ × 3 × 101 = 9696 

(iv) Prime factorization:

144 = 2⁴ × 3² 

198 = 2 × 3² × 11 

HCF = product of smallest power of each common prime factor in the numbers 

= 2 × 3² = 18 

LCM = product of greatest power of each prime factor involved in the numbers 

= 2⁴ × 3² × 11 = 1584 

(v) Prime factorization: 

396 = 2² × 3² × 11 

1080 = 2³ × 3³ × 5 

HCF = product of smallest power of each common prime factor in the numbers 

= 2² × 3² = 36 

LCM = product of greatest power of each prime factor involved in the numbers 

= 2³ × 3³ × 5 × 11 = 11880 

(vi) Prime factorization: 

1152 = 2⁷ × 3² 

1664 = 2⁷ × 13 

HCF = product of smallest power of each common prime factor in the numbers 

= 2⁷ = 128 

LCM = product of greatest power of each prime factor involved in the numbers = 2⁷× 3² × 13 = 14976