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.
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