H and L are the HCF and LCM respectively of two numbers \(x\) and \(y\). If \(H+L=x+y\), what is the value of \(H^{3}+L^{3}\)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(x^{3}+y^{3}\)
For two numbers, \(HL=xy\) (HCF \(\times\) LCM \(=\) product). Two pairs with the same sum and the same product are the same pair, so \(H\) and \(L\) are \(x\) and \(y\) in some order.
\(H+L=x+y\) (given) and \(HL=xy\)
\(H^{3}+L^{3}=(H+L)^{3}-3HL(H+L)\)
\(=(x+y)^{3}-3xy(x+y)\)
\(=x^{3}+y^{3}\)
So \(H^{3}+L^{3}=\) \(x^{3}+y^{3}\).