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Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.

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Best answer

Let the positive integer = a

According to Euclid’s division lemma,

a = bm + r

According to the question, b = 5

a = 5m + r

So, r= 0, 1, 2, 3, 4

When r = 0, a = 5m.

When r = 1, a = 5m + 1.

When r = 2, a = 5m + 2.

When r = 3, a = 5m + 3.

When r = 4, a = 5m + 4.

Now,

When a = 5m

a2 = (5m)2 = 25m2

a2 = 5(5m2) = 5q, where q = 5m2

When a = 5m + 1

a2 = (5m + 1)2 = 25m2 + 10 m + 1

a2 = 5 (5m2 + 2m) + 1 = 5q + 1, where q = 5m2 + 2m

When a = 5m + 2

a2 = (5m + 2)2

a2 = 25m2 + 20m + 4

a2 = 5 (5m2 + 4m) + 4

a2 = 5q + 4 where q = 5m2 + 4m

When a = 5m + 3

a2 = (5m + 3)2 = 25m2 + 30m + 9

a2 = 5 (5m2 + 6m + 1) + 4

a2 = 5q + 4 where q = 5m2 + 6m + 1

When a = 5m + 4

a2 = (5m + 4)2 = 25m2 + 40m + 16

a= 5 (5m2 + 8m + 3) + 1

a= 5q + 1 where q = 5m2 + 8m + 3

Therefore, square of any positive integer cannot be of the form 5q + 2 or 5q + 3.

Hence Proved.

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