At a Republic Day march-past, a band of 48 members marches ahead of an army contingent of 312 members. Both must march in the same number of columns, as many as possible. How many rows will the army contingent form?

**Theorem**: A rational number $\frac{p}{q}$ (in lowest terms) has a terminating decimal expansion if and only if $q$ has the form **2ᵐ × 5ⁿ** (m, n ≥ 0).

**Examples**:
- $\frac{3}{8}:q=8=2^3$ → terminates$.(\frac{3}{8}=\mathrm{0.375.})-\frac{7}{40}:q=40=2^3\times 5$ → terminates$.(\frac{7}{40}=\mathrm{0.175.})-\frac{1}{3}:q=3$ → does NOT terminate$.(\frac{1}{3}=0.333$… recurring.)
- $\frac{4}{15}:q=15=3\times 5$ → does NOT terminate (because of the 3).

**Why**: any decimal expansion is a fraction with denominator 10ⁿ = 2ⁿ × 5ⁿ. To convert $\frac{p}{q}$ to decimal form, you need $q$ to ONLY have 2s and 5s as primes — otherwise you can't multiply numerator/denominator into a power of 10.