Question

1. Show that any positive even integer is of the form4q,or4q+2,where q is some integer
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Answer 1

AnswEr:

$\bold{\underline{\sf{\red{By,\: Using\: Euclid's\: Division\:Lemma}}}}$

Let us Consider that a & b be are any two positive integers.

$:\implies\sf\: a = bq + r$

Here, b = 4

$:\implies\sf\:a = 4q + r$

r can be 0, 1, 2 & 3

$\rule{150}2$

If r = 0

$:\implies\sf\:a = 4q$

If r = 1

$:\implies\sf\: a = 4q + 1$

if r = 2

$:\implies\sf\: a = 4q + 2$

If r = 3

$:\implies\sf\: a = 4q + 3$

Therefore, a = 4q, 4q + 2 are even integers.

And, a = 4q + 1, 4q + 3 are odd integers.

Hence, we can say that any positive even integers is in the form of 4q, 4q + 2.

Hence, Proved!

$\rule{150}2$

Answer 2

Hey!!

Solution:

We need to consider any positive even integer as 'a'.

On dividing a by 4 , let 'q' be quotient and 'r' be the remainder.

Now, using Euclid's Division lemma, we have

a = 4q + r where r = 0, 1, 2, 3 ie,

a = 4q + 0 = 4q

a = 4q + 1

a = 4q + 2

a = 4q + 3

But a = 4q + 1 and a = 4q + 3 are odd values of a.

Therefore, a being an odd integer, we have, a = 4q or a = 4q + 2.