Using Euclid Division Lemma to prove the given statement

Euclid division lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < />

Proof:

Let's assume that n is a positive integer and it can be expressed as 5m + 2 or 5m + 3, where m is an integer.

Case 1: n = 5m + 2
Using Euclid division lemma, we can write n as:
n = 5m + 2 = 5m + 5 - 3
n = 5(m + 1) - 3

Now, let's consider the square of n:
n^2 = (5(m + 1) - 3)^2
n^2 = 25(m + 1)^2 - 30(m + 1) + 9
n^2 = 5(5(m + 1)^2 - 6(m + 1)) + 9

We can see that the square of n is of the form 5q + 9, where q is an integer. Therefore, n^2 cannot be of the form 5m + 2.

Case 2: n = 5m + 3
Using Euclid division lemma, we can write n as:
n = 5m + 3 = 5m + 5 - 2
n = 5(m + 1) - 2

Now, let's consider the square of n:
n^2 = (5(m + 1) - 2)^2
n^2 = 25(m + 1)^2 - 20(m + 1) + 4
n^2 = 5(5(m + 1)^2 - 4(m + 1)) + 4

We can see that the square of n is of the form 5q + 4, where q is an integer. Therefore, n^2 cannot be of the form 5m + 3.

Conclusion:

Hence, we have proved that the square of any positive integer cannot be of the form 5m + 2 or 5m + 3 for some integer m using Euclid division lemma.