Exactly one of these numbers has an ODD number of positive divisors. Which one?

Divisors normally come in pairs, $d$ with $n÷d$, so the count is even — unless one divisor pairs with itself, which happens only when the number is a perfect square. Of the four, only $196=14^2$ is a square, and $196=2^2\times 7^2$ gives $(2+1)(2+1)=9$ divisors. The others come out even: $48=2^4\times 3$ has 10 divisors, $72=2^3\times 3^2$ has 12 and $250=2\times 5^3$ has 8.