CEMC Fermat

Algebra Difficulty 2.5 Junior Find the answer

For some integers $m$ and $n$ , the expression $(x+m)(x+n)$ is equal to a quadratic expression in $x$ with a constant term of -12. Which of the following cannot be a value of $m$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Expanding, $(x+m)(x+n)=x^{2}+n x+m x+m n=x^{2}+(m+n) x+m n$ . The constant term of this quadratic expression is $m n$ , and so $m n=-12$ . Since $m$ and $n$ are integers, they are each divisors of -12 and thus of 12. Of the given possibilities, only 5 is not a divisor of 12, and so $m$ cannot equal 5.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.