TMUA 2018 Paper 2 Q17 — Worked Solution

P2·Number Theory & Divisibility, Logic & Quantifiers·6.0Official · 2018 · Q17

fermat-two-squarescounterexampleprime-factorisation

A positive integer is called a squaresum if and only if it can be written as the sum of the squares of two integers. For example,

61

and

9

are both squaresums since

61 = 5^{2} + 6^{2}

and

9 = 3^{2} + 0^{2}

.A prime number is called awkward if and only if it has a remainder of

3

when divided by

4

. For example,

23

is awkward since

23 = 5 \times 4 + 3

.A (true) theorem due to Fermat states that:A positive integer is a squaresum if and only if each of its awkward prime factors occurs to an even power in its prime factorisation.It follows that

5 \times 2 3^{2}

is a squaresum, since

23

occurs to the power

2

, but

5 \times 2 3^{3}

is not, since

23

occurs to the power

3

.Which one of the following statements is not true?

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TMUA 2018 Paper 2 Question 17

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