Question

Prove that every odd integer is congruent to 1 modulo 4 or to 3 modulo 4.

Short answer

It is proved that every odd integer is congruent to 1 modulo 4 or to 3 modulo 4.

Step-by-step solution

Definition of Congruence

Consider a, b, n as an integer with $n>0$. If n divides $a-b$ then a iscongruent tob modulo n (expressed as $a\equiv b\left(modn\right)$).

Show that every odd integer is congruent to 1 modulo 4 or to 3 modulo 4

Every number $a\in \mathrm{\mathbb{Z}}$ would be of the form $4k,4k+1,4k+2,4k+3$. They have been even numbers because $2|4k$and $2|\left(4k+2\right)$.

The odd numbers are $4k+3$and $4k+1$; then

$4k+1\equiv 1\left(mod4\right)$

$4k+3\equiv 3\left(mod4\right)$

As a result, every odd number is congruent to either 1 modulo 4 or 3 modulo 4.

Thus, it is proved that every odd integeris congruent to 1 modulo 4 or to 3 modulo 4.