2024 AMC 10A Problem 7

A positive product uses either three positive factors or one positive and two negative factors

Two negative factors let the positive factor be large; minimize the positive factor minus the sum of the two magnitudes

A positive product comes from three positive integers or one positive and two negative integers. Three positive integers have sum at least $3.$ In the second case write the numbers as $-x,-y,z,$ where $x,y,z$ are positive and $xyz=60.$ A positive sum requires $z \gt x+y \ge 2\sqrt{xy},$ so $\frac{60}{xy} \gt 2\sqrt{xy}$ and hence $xy \lt 10.$ The possible products $xy$ that divide $60$ are $1,2,3,4,5,6.$ Checking their factor pairs, the smallest positive value of $\frac{60}{xy}-x-y$ is $10-1-6=3.$ Thus $(-1)(-6)(10)=60,$ and no positive sum below $3$ is possible. Therefore, the answer is B.