Question

5. Show that the square of any positive integer is of the form 2m or 2m + 1 for some
integer m, using Euclid's division lemma.​

Answer 1

Answer:

Hey Mate ✌

Here's your answer friend,

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==> Let a be the positive integer for some integer q.

==> We know every positive integer is in the form of 2q, or 2q + 1.

==> Here, b = 2,

==> Therefore, r = 0,1 .........{ 0>=r>b}

==> By using Euclid's division lemma,

we get,

For r = 0

==> a = bq + r

==> a = 2q + 0

==> a = 2q

==> square of positive integer

==> a² = (2q)²

==> a² = 4q²

==> a² = 2(2q²)

==> a² = 2m .........{ where m = 2q²}

Now,

when r = 1,

we get,

==> a = bq + r

==> a = 2q + 1

==> Square of positive integer

==> a² = (2q + 1)²

==> a² = 4q² + 4q + 1

==> a² = 2(2q² + 2q) + 1

==> a² = 2m + 1 ............{ where m = 2q² + 2q}

Hence, showed square of any positive integer is either in the form of 2m or 2m + 1.

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⏩ Hope it helps you ^_^