The LCM of two numbers \(a\) and \(b\) is 204 times their HCF. If their HCF is 12 and the difference between the numbers is 60, then what is \(a+b\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 348
Two numbers with HCF 12 are \(12m\) and \(12n\), where \(m\) and \(n\) have no common factor; their LCM is \(12mn\). The given ratio of LCM to HCF fixes \(mn\), and the difference picks the pair.
LCM \(\div\) HCF \(=mn=204\)
Difference: \(12(m-n)=60\), so \(m-n=5\)
Co-prime pairs with product 204: (1, 204), (3, 68), (4, 51), (12, 17)
Only (12, 17) has a difference of 5
Numbers \(=12\times12=144\) and \(12\times17=204\)
\(a+b=144+204=348\)
So \(a+b=\) 348.