Question

If two positive integers p and q are written as p=a^2b^3 and q=a^3b, a and b are prime numbers, then verify LCM(p,q)×HCF (p,q)=pq.

Answer 1

$p = {a}^{2} {b}^{3} \\ q = {a}^{3}b \\ lcm(p \: q ) = {a}^{3} {b}^{3} \\ hcf(p \: q) = {a}^{2} b \\ lcm(p \: q) \times hcf(p \: q) = {a}^{3} {b}^{3} \times {a}^{2} b \\ = {a}^{5} {b}^{4} \: \: \: \: \: eq1 \\ p \times q = {a}^{2} { b}^{3} \times {a}^{3} b \\ = {a}^{5} {b}^{4} \: \: \: eq2 \\ eq1 = eq2 \\ hence \: prove$