The LCM of two numbers \(a\) and \(b\) is 28 times their HCF. If their HCF is 15 and the difference between the numbers is 45, what is \(a+b\)?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 165
Write the numbers as \(15 x\) and \(15 y\) with \(x\), \(y\) co-prime; then the LCM is \(15 xy\), so \(xy=28\).
Co-prime pairs with product 28: (7, 4), (28, 1)
Difference \(15(x-y)=45\), so \(x-y=3\); only (7, 4) works
Numbers: 105 and 60; sum \(=165\)
So \(a+b=\) 165.
Why the other options are incorrect
Option (2) 435 — Incorrect. This uses the co-prime pair 28 and 1, whose numbers 420 and 15 differ by 405, not 45.
Option (3) 11 — Incorrect. This is \(7+4\); it still has to be multiplied by the HCF 15.
Option (4) 420 — Incorrect. This is the LCM, 420, not the sum of the numbers.