# A question on Remainder Theorem

Is there a rational polynomial$f(x)$ of degree 100 such that its remainder after division by $g(x)=x^{60} -2$ and $h(x)=x^{49} +3$ are the same?

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Clearly(?), $g$ and $h$ are relatively prime, therefore $f\equiv p\pmod g$ and $f\equiv p\pmod h$ if and only if $f\equiv p\pmod{gh}$. Thus if $f$ is a ploynomial with same remainder $p$ after dividing by $g$ and by $h$, then $f-p$ is a multiple of $x^{109}+\ldots$. As $\deg p<\deg g=49$ we can only have $f=p$ or $\deg f\ge 109$, hence definitely $\deg f\ne 100$.