OCR Further Additional Pure AS 2021 November — Question 7

Exam BoardOCR
ModuleFurther Additional Pure AS (Further Additional Pure AS)
Year2021
SessionNovember
TopicNumber Theory

7
  1. Let \(f ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }\). Use arithmetic modulo 11 to prove that \(\mathrm { f } ( n ) \equiv 0 ( \bmod 11 )\) for all integers \(n \geqslant 0\).
  2. Use the standard test for divisibility by 11 to prove the following statements.
    1. \(10 ^ { 33 } + 1\) is divisible by 11
    2. \(10 ^ { 33 } + 1\) is divisible by 121