Question

Prove that square of any positive integer is of the form 5m+1 will leave a remainder 1 when divided by 5 for some integer m.

Answer 1

let n is any +ve integer
By ED

Answer 2

To prove:

Square of 5m+1 leaves 1 as reminder for all the positive integers when divided by 5.

Solution:

$n = (5m + 1)^2$

$n = 25m^2 + 1 + 10m \rightarrow (1)$

Let us assume a value for m. The value should be a positive integer.

Thus, let us take m = 2

Now, on substituting value of m in equation (1), we get,

$n = 25(1)^2 + 1 + 10(1)$

On simplifying, we get,

n = 25 + 1 + 10

Thus,

n = 36

Now, on dividing the value of n with 5, we get,

$n = \frac {36}{5}$

We get the reminder as 1

Thus, $(5m + 1)^2$ gets 1 as reminder for any positive integer.

Hence proved.