Let \(X\) be the greatest number which, when it divides 1309, 2384 and 4190, leaves the same remainder \(Y\) in each case. What is the value of \(X-Y\)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 24
If the remainders are equal, the differences of the numbers are divisible exactly by the divisor, so the greatest divisor is the HCF of the differences.
Differences: \(2384-1309=1075\), \(4190-2384=1806\), \(4190-1309=2881\)
\(1075=5^{2}\times43\), \(1806=2\times3\times7\times43\), \(2881=43\times67\)
HCF \(=X=43\)
\(1309=43\times30+19\), so \(Y=19\)
\(X-Y=43-19=24\)
The value of \(X-Y\) is 24.
Why the other options are incorrect
Option (1) 67 — Incorrect. This takes \(X=86\), but 86 does not divide the differences exactly (it does not divide 1075).
Option (2) 62 — Incorrect. This is \(X+Y=43+19\), not \(X-Y\).
Option (3) 43 — Incorrect. This is \(X\) alone; the remainder 19 still has to be taken away.