Challenging Questions

Mathematics – Compulsory Part

Learning Unit: More about polynomials

Question M0-04-08

Consider \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0 \).

1. Find the remainder when \( f(x) \) is divided by \( (x – 1) \).
2. If \( f(10) = 12345678 \), find the possible values of \( a_n, a_{n-1}, \dots, a_1, a_0 \).
3. Hence, show that if the sum of digits of a positive integer is divisible by 9, then the integer is also divisible by 9.
4. Using a similar argument, show that if the difference between the sums of the digits in odd and even positions of a positive integer is divisible by 11, then the integer is also divisible by 11.

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