What is the least number of soldiers who can be arranged in 8, 12, 16 or 20 rows with an equal number in each row, and can also be arranged in the form of a square?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 3600
The number must be a multiple of the LCM of 8, 12, 16 and 20 and also a perfect square. A number is a perfect square when every prime in it has an even power.
\(8=2^{3}\), \(12=2^{2}\times3\), \(16=2^{4}\), \(20=2^{2}\times5\)
LCM \(=2^{4}\times3\times5=240\)
3 and 5 have odd powers, so multiply by \(3\times5=15\)
\(240\times15=3600=60^{2}\)
The least number of soldiers is 3600.
Why the other options are incorrect
Option (1) 240 — Incorrect. It fills all the rows, but it is not a perfect square.
Option (2) 2500 — Incorrect. It is \(50^{2}\), but it is not divisible by 8.
Option (4) 6400 — Incorrect. It is \(80^{2}\), but it is not divisible by 12.