A man has some bananas. If he gives 6 or 8 bananas to each person, 4 bananas are left over. If he gives 10 or 12 bananas to each person, 4 bananas are left over this time also. What is the minimum number of bananas he has?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 124
The same 4 bananas are left over in every case, so the number of bananas minus 4 must be divisible by 6, 8, 10 and 12. The number is therefore a multiple of the LCM plus 4.
\(6=2\times3\), \(8=2^{3}\), \(10=2\times5\), \(12=2^{2}\times3\)
LCM \(=2^{3}\times3\times5=120\)
Number of bananas \(=120k+4\)
At least one person gets bananas, so \(k\) is at least 1, giving \(120+4=124\)
The minimum number of bananas is 124.
Why the other options are incorrect
Option (1) 484 — Incorrect. It is \(120\times4+4\), so it fits the conditions, but it is not the minimum.
Option (2) 304 — Incorrect. \(304-4=300\) is not divisible by 8.
Option (4) 260 — Incorrect. \(260-4=256\) is not divisible by 6.