UPSC 2017 CDS I Math Paper

Let a = 6, b = 2 and c = 3
Here, a = bc and HCF (b, c) = 1
Now, c and d can be co-prime or one is the multiple of the other. So, there are two cases.
Take d = 8.
HCF(c, d) = HCF(3, 8) = 1
HCF(c, bd) = HCF(3, 16) = 1
So, HCF(c, bd) = HCF(c, d)
Take d = 9.
HCF(c, d) = HCF(3, 9) = 3
HCF(c, bd) = HCF(3, 18) = 3
So, HCF(c, bd) = HCF(c, d)
Hence, statement 1 is correct.
Take d = 8.
LCM(a, d) = LCM(6, 8) = 24
LCM(c, bd) = LCM(3, 16) = 38
So, LCM(a, d) ≠ LCM(c, bd)
Take d = 9.
LCM(a, d) = LCM(6, 9) = 18
LCM(c, bd) = LCM(3, 18) = 18
So, LCM(a, d) = LCM(c, bd)
Hence, statement 2 is not correct.