Question

An army contingent of 104 members is to march behind an army band of 96 members in a parade.

Answer 1

You can dispose a number $x$ of elements in a matrix-like formation with $n\times m$ shape if and only if $n$ and $m$ both divide $x$, and also $nm=x$.

So, we need to find the greatest common divisor between $104$ and $96$, so that we can use that divisor as the number of columns, and then.

To do so, we need to find the prime factorization of the two numbers:

$104 = 2^3\times 13$

$96 = 2^5 \times 3$

So, the two numbers share only one prime in their factorization, namely $2$, but we can't take "too many" of them: $104$ has "three two's" inside, while $96$ has "five two's" inside. So, we can take at most "three two's" to make sure that it is a common divisor. As for the other primes, we can't include $3$ nor $13$, because it's not a shared prime.

So, the greater number of columns is $2^3=8$, which yield the following formations:

$104 \to 8\times 13$

$96 \to 8\times 12$