Mathematics A, Further H235: Number Theory Topical Past Paper Questions

For positive integers $n$, let $f(n) = 1 + 2^{n} + 4^{n}$. (a) (i) Given that $n$ is a multiple of 3, but not of 9, use the division algorithm to write down the two possible forms that $n$ can take. [1] (ii) Show that when n is a multiple of 3, but not of 9, f(n) is a multiple of…

(a) Express as a decimal (base-10) number the base-23 number $ 7119_{23} $. (b) Solve the linear congruence $ 7n+11 \equiv 9(\mod 23) $. [3] [3] (c) Let $N = 10a + b$ and $M = a + 7b$, where $a$ and $b$ are integers and $0 \leq b \leq 9$. (i) By considering $3N-7M$, prove that $2…

(a) Let $ f(n) = 2^{4n+3} + 3^{3n+1} $. Use arithmetic modulo 11 to prove that $ f(n) \equiv 0 \pmod{11} $ for all integers $ n \geqslant 0 $. [4] b) Use the standard test for divisibility by 11 to prove the following statements. (i) $ 10^{33} + 1 $ is divisible by 11 (ii) $ 10^{…