Composite number proofs

Questions requiring proof that an expression is always composite (not prime) by factorization or other methods.

3 questions · Challenging +1.6

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OCR Further Additional Pure 2023 June Q8
9 marks Challenging +1.8
8 Let \(f ( n )\) denote the base- \(n\) number \(2121 _ { n }\) where \(n \geqslant 3\).
    1. For each \(n \geqslant 3\), show that \(\mathrm { f } ( n )\) can be written as the product of two positive integers greater than \(1 , \mathrm { a } ( n )\) and \(\mathrm { b } ( n )\), each of which is a function of \(n\).
    2. Deduce that \(\mathrm { f } ( n )\) is always composite.
  2. Let \(h\) be the highest common factor of \(\mathrm { a } ( n )\) and \(\mathrm { b } ( n )\).
    1. Prove that \(h\) is either 1 or 5 .
    2. Find a value of \(n\) for which \(h = 5\).
OCR Further Additional Pure AS 2017 December Q5
7 marks Challenging +1.8
5 Given that \(n\) is a positive integer greater than 2 , prove that
  1. \(\quad 10201 _ { n }\) is a square number.
  2. \(\quad 1221 _ { n }\) is a composite number.
OCR Further Additional Pure 2018 September Q1
5 marks Challenging +1.2
1
  1. Write the number \(100011 _ { n }\), where \(n \geqslant 2\), as a polynomial in \(n\).
  2. Show that \(n ^ { 2 } + n + 1\) is a factor of this expression.
  3. Hence show that \(100011 _ { n }\) is composite in any number base \(n \geqslant 2\).