1977 Canadian MO Problems/Problem 4


Let \[p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0\] and \[q(x)=b_mx^m+b_{m-1}x^{m-1}+\cdots +b_1x+b_0\] be two polynomials with integer coefficients. Suppose that all of the coefficients of the product $p(x)\cdot q(x)$ are even, but not all of them are divisible by 4. Show that one of $p(x)$ and $q(x)$ has all even coefficients and the other has at least one odd coefficient.


suppose p(x) and q(x) have all even coefficient then as p(x).q(x) involves all the ai(x).bj(x).. all the coefficient of p(x).q(x) must be divisible by 4. which is certainly not the case, hence p(x) or q(x) has atleast one odd coefficient. Without the loss of generality assume p(x) has an odd coefficient ak(x)^l if q(x) also has an odd coefficient say bt(x)^r the p(x).q(x) will involve the term ak*bt(x)^r+k.. and as both are odd the coefficient is also odd... hence contradiction sorry ,i dont know latex, so writing in this manner