Question

Show that any positive even integer is in the form of 4q, 4q+2, where q is some integer

Answer 1

Let "a" and "b" be any positive integer.

Now according to Euclid's Division Lemma

there exists unique integers q and r satisfying

a=bq+r   where 0≤r<b.

Here, b=4

Since,0≤r<4 ,a can be 4q , 4q+1 , 4q+2 , 4q+3

But, 4q+1 and 4q+3 are odd numbers so we can't count it now,

So, a can be 4q, 4q+2

Thus any positive even integer can be written in the form 4q , 4q+2 , where q is an integer.

Thanks!!!

Answer 2

Step-by-step explanation:

Let a be the positive integer.

And, b = 4 .

Then by Euclid's division lemma,

We can write a = 4q + r ,for some integer q and 0 ≤ r < 4 .

°•° Then, possible values of r is 0, 1, 2, 3 .

Taking r = 0 .

→ a = 4q .

Taking r = 1 .

→ a = 4q + 1 .

Taking r = 2

→ a = 4q + 2 .

Taking r = 3 .

→ a = 4q + 3 .

But a is an even positive integer, so a can't be 4q + 1 , or 4q + 3 [ As these are odd ] .

∴ a can be of the form 4q or 4q + 2 for some integer q .

Hence , it is solved