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 = 52 + 62 and 9 = 32 + 02.
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× 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× 232 is a squaresum, since 23 occurs to the power 2, but 5× 233 is not, since 23 occurs to the power 3.
Which one of the following statements is not true?
- AEvery square number is a squaresum.
- BIf N and M are squaresums, then so is NM.
- CIf NM is a squaresum, then N and M are squaresums.
- DIf N is not a squaresum, then kN is a squaresum for some number k which is a product of awkward primes.
Show the answer and worked solution
answer · C
- AEvery square number is a squaresum.
- BIf N and M are squaresums, then so is NM.
- CIf NM is a squaresum, then N and M are squaresums.
- DIf N is not a squaresum, then kN is a squaresum for some number k which is a product of awkward primes.
Read every option through the theorem, as a statement about the parity of the exponents of the awkward primes. A square number has every exponent even, so the first is true. Multiplying adds exponents, so even plus even stays even and the second is true. For the last, take the awkward primes appearing to an odd power in N and let k be their product: each of those exponents goes up by one and becomes even, so kN is a squaresum. The third fails: 3× 3 = 9 is a squaresum, but 3 on its own is not.