What is the smallest positive integer m such that the following equation
holds for some polynomials u(x) and v(x) with integer coefficients?
(x + 2)(x + 5)(x + 7) u(x) − (x − 2)(x − 5)(x − 7) v(x) = m  Answers.
A) 70 B) 1260 C) 1680 D) 3780 E) 7560
Substituting x = 5 into the given equation, we deduce 7 · 10 · 12 · u(5) = m. Substituting x = 7   into the given equation, we deduce 9 · 12 · 14 · u(7) = m. It follows that m must be divisible by   each of 7 · 10 · 12 and 9 · 12 · 14 which implies that 23 · 33 · 5 · 7 = 7560 divides m. Of the choices,   we see that only (e) is possible here. Verifying that such a u(x) and v(x) exist requires a little more   work, but we note that one can take u(x) = x2 − 14x + 54 and v(x) = x2 + 14x + 54.   correct answer 7560
<result>E</result>
What is the smallest positive integer m such that the following equation
holds for some polynomials u(x) and v(x) with integer coefficients?
(x + 2)(x + 5)(x + 7) u(x) − (x − 2)(x − 5)(x − 7) v(x) = m