Question

use Euclid division lemma to show that the square of any postive integer cannot be of the form 5m + 2 or 5m + 3 for some integer m.​

Answer 1

Answer:

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Answer 2

Let:-

n be the postive number.

By Euclid's division lemma

n = 5q + r [0 ≤ r < 5]

Then:-

n = 5q,

n = 5q + 1

n = 5q + 2

n = 5q + 3

n = 5q + 4

where, q is a whole number.

Now:-

=> n² = (5q)²

=> n² = 25q²

=> n² = 5(5q²)

=> n² = 5m

=> n² = (5q + 1)²

=> n² = 25q² + 10q + 1

=> n² = 5m + 1

=> n² = (5q + 2)²

=> n² = 25q² + 20q + 4

=> n² = 5m + 4

Similarly:-

=> n² = (5q + 3)²

=> n² = 25q² + 30q + 5 + 4

=> n² = 5m + 4

And,

n² = (5q + 4)² = 5m + 1

Therefore:-

We can say that square of any positive integer cannot be in the form of 5m + 2 or 5m + 3.