(a) Show that a square number $a^2$ divided by 3 gives only remainders 0 or 1.
(b) Show that a square number $a^2$ divided by 4 gives only remainders 0 or 1.
(a) We start by noting that a number $a$ is congruent to 0, 1 or 2 modulo 3.
Using Proposition (3.8) we have, for each of these cases
$$ a \equiv 0 \pmod 3 \implies a^2 \equiv 0^2 \equiv 0 \pmod 3 $$
$$ a \equiv 1 \pmod 3 \implies a^2 \equiv 1^2 \equiv 1 \pmod 3 $$
$$ a \equiv 2 \pmod 3 \implies a^2 \equiv 2^2 \equiv 1 \pmod 3 $$
And so $a^2$ divided by 3 only gives remainders 0 or 1.
(b) We start by noting that a number $a$ is congruent to 0, 1, 2 or 3 modulo 4.
Using Proposition (3.8) we have, for each of these cases
$$ a \equiv 0 \pmod 4 \implies a^2 \equiv 0^2 \equiv 0 \pmod 4 $$
$$ a \equiv 1 \pmod 4 \implies a^2 \equiv 1^2 \equiv 1 \pmod 4 $$
$$ a \equiv 2 \pmod 4 \implies a^2 \equiv 2^2 \equiv 0 \pmod 4 $$
$$ a \equiv 3 \pmod 4 \implies a^2 \equiv 3^2 \equiv 1 \pmod 4 $$
And so $a^2$ divided by 4 only gives remainders 0 or 1.