If two integers p and q are in the ratio 4: 5 and their least common

LCM (p,q) = \(2^2 * 2^3 * 5^2\) (I have purposefully written \(2^2\) and \(2^3\) separately)

Given \(\frac{p }{ q}\) = \(\frac{2^2 }{ 5}\)

This means, there was certain factor that was common both in p and q and that got cancelled out in the ratio.

Let's see if we can construct that factor -

LCM is obtained by considering the maximum power of the prime factors in p and q, therefore the missing \(2^3\) and 5 must be the common factor that got cancelled out when p was divided by q. Note that this factor is the greatest common factor, as any higher power of either 2 or of 5 in the common factor must be accounted in the LCM, and thus will change the value of the LCM.

Hence GCD = \(2^3 * 5\) = 40

We can cross verify \(p = 2^2 * 2^3 * 5 ; q = 2^3 * 5^2\)

\(\frac{p}{q}\) = \(\frac{2^2 * 2^3 * 5 }{ 5 * 2^3 * 5 }\)

HCF of \(p = 2^2 * 2^3 * 5 ; q = 2^3 * 5^2 = 40\)

Option B