Prove that the square of any positive integer of the form $5q+1$ is of the same form.

Given: Statement "Square of any positive integer of the form $5q\ +\ 1$ is of the same form".

To prove: Here we have to prove the given statement.

Solution:

According to Euclid's lemma,

If $a$ and $b$ are two positive integers;

• $a\ =\ bq\ +\ r$, where $0\ \underline{< }\ r\ <\ b$.

Let, $a\ =\ 5q\ +\ 1$, where $b\ =\ 5$ and $r\ =\ 1$

Now,

$a\ =\ 5q\ +\ 1$

Squaring both sides, we get:

$a^2\ =\ (5q\ +\ 1)^2$

$a^2\ =\ 25q^2\ +\ 10q\ +\ 1$

$a^2\ =\ 5(5q^2\ +\ 2q)\ +\ 1$

$a^2\ =\ 5m\ +\ 1$, where $m\ =\ 5q^2\ +\ 2q$

Hence, the square of any positive integer of the form $5q\ +\ 1$ is of the same form.

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

27 Views

Kickstart Your Career

Get certified by completing the course

Get Started