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(C) For any polynomial $f(x)$ with integer coefficients,the property $(a-b)$ divides $(f(a)-f(b))$ holds for any distinct integers $a$ and $b$.Given $

Vedclass · Accepted Answer

$27$

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(C) For any polynomial $f(x)$ with integer coefficients,the property $(a-b)$ divides $(f(a)-f(b))$ holds for any distinct integers $a$ and $b$.Given $f(1)=5$ and $f(2)=7$,we have $(2-1)$ divides $(f(2)-f(1))$,which is $1$ divides $(7-5)=2$. This is always true.For $f(12)$,we have $(12-1)$ divides $(f(12)-f(1)) \implies 11$ divides $(f(12)-5)$.Also,$(12-2)$ divides $(f(12)-f(2)) \implies 10$ divides $(f(12)-7)$.Let $f(12) = k$. Then $k \equiv 5 \pmod{11}$ and $k \equiv 7 \pmod{10}$.From $k \equiv 7 \pmod{10}$,$k$ can be $7, 17, 27, 37, \dots$.Checking these values for $k \equiv 5 \pmod{11}$:$7 \equiv 7 \pmod{11}$$17 \equiv 6 \pmod{11}$$27 \equiv 5 \pmod{11}$.Thus,the smallest positive value is $27$.

Let $f(x)$ be a polynomial with integer coefficients satisfying $f(1)=5$ and $f(2)=7$. The smallest possible positive value of $f(12)$ is

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