Proof by exhaustion with table

A question is this type if and only if it provides a table to fill in and asks to prove a statement by exhaustively checking all valid combinations of constrained integer variables (e.g., a + b + c = 10 with c = b + 2).

3 questions · Easy -1.4

1.01a Proof: structure of mathematical proof and logical steps
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Edexcel P2 2022 October Q1
3 marks Easy -1.2
  1. Given that \(a , b\) and \(c\) are integers greater than 0 such that
  • \(c = b + 2\)
  • \(a + b + c = 10\)
Prove, by exhaustion, that the product of \(a , b\) and \(c\) is always even.
You may use the table below to illustrate your answer. You may not need to use all rows of this table.
\(a\)\(b\)\(c\)
1
2
Edexcel P2 2023 October Q1
3 marks Easy -1.2
  1. Given that \(a , b\) and \(c\) are integers greater than 0 such that
  • \(c = 3 a + 1\)
  • \(a + b + c = 15\) prove, by exhaustion, that the product \(a b c\) is always a multiple of 4
    You may use the table below to illustrate your answer.
You may not need to use all rows of this table.
\(a\)\(b\)\(c\)\(a b c\)
WJEC Unit 1 2019 June Q05
3 marks Easy -1.8
Given that \(n\) is an integer such that \(1 \leq n \leq 4\), prove that \(2n^2 + 5\) is a prime number. [3]