Competition problem — Algebra — Maths Olympiad Prep

Algebra Difficulty 4.9 AIME Prove it Brazil

The remainder on dividing the polynomial $p(x)$ by $x^2 - (a+b)x + ab$ (where $a$ and $b$ are unequal) is $mx + n$ . Find the coefficients $m, n$ in terms of $a, b$ . Find $m, n$ for the case $p(x) = x^{200}$ divided by $x^2 - x - 2$ and show that they are integers.

Solution

Let $p(x) = q(x)(x-a)(x-b) + mx + n$ . So putting $x = a, b$ we get $p(a) = ma + n$ , $p(b) = mb + n$ . Solving, $m = \frac{p(a)-p(b)}{a-b}$ , $n = \frac{p(b)a-p(a)b}{a-b}$ .

In the case given $a = 2$ , $b = -1$ , so $m = \frac{2^{200}-1}{3}$ , $n = \frac{2^{200}+2}{3}$ . Note that $2 \equiv -1 \pmod{3}$ , so $2^{200} \equiv 1 \pmod{3}$ and hence $2^{200} - 1$ is a multiple of $3$ , so $m$ is an integer. $n = m + 1$ , so $n$ is also an integer.

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