The HCF h of a and b is doubled when the first number is doubled and HCF is tripled when the second number is tripled. The HCF of which two below given numbers is 6h ?

Correct option is D

Given:

HCF(a, b) = h; HCF(2a, b) = 2h; HCF(a, 3b) = 3h.

Formula Used:
Let a=hm,  b=hn with gcd(m,n)=1.
gcd(ku,kv) =k gcd(u,v); use properties of gcd.

Solution:
From HCF(2a , b) = 2h

gcd(2m , n) = 2

From HCF(a , 3b) = 3h

gcd(m , 3n) = 3

Write m=$$3m_1$$​ and n$$=2n_1$$​ with gcd$$(m_1,n_1)$$​=1. Then

HCF(2a,3b)=$$h \cdot gcd(2m,3n)=h\cdot \gcd(6m_1,6n_1)=6h.$$​

The pair is 2a and 3b (their HCF is 6h).