tmua.fyi Paper 2 Challenge 2 · Question 8

Consider the claim: if $3$ divides $a^2+b^2$ then $3$ divides $a$ and $3$ divides $b$ , where $a$ and $b$ are integers. Which of the following is correct?

ATrue: a square is congruent to

0

1

modulo

3

, so a sum of two squares is divisible by

3

only when both squares are.

BTrue, but only when

a

and

b

are both positive.

CFalse, as

a=1,b=2

shows.

DFalse, as

a=3,b=6

shows.

ETrue, and it follows at once from the fact that

3

is prime, with no further argument needed.

FFalse, as

a=1,b=5

shows.

GTrue, because

a^2+b^2=(a+b)^2-2ab

and

3

must therefore divide

2ab

HFalse, as

a=2,b=4

shows.

examday

tmua.fyi Paper 2 Challenge 2 · Q8