Question

Show that any positive even integer is of the form 4q or 4q+2 where q is a whole number

Answer 1

To show: Any positive integer is of the form 4q or 4q + 2 for any whole number q.

Procedure:

=> Write first a few positive even integers.

2, 4, 6, 8, 10, 12,...

=> Divide each integer by 4 and write the remainders thus obtained.

2, 0, 2, 0, 2, 0,...

Observation:

The sequence of the remainders is the repetition of the sequence 2, 0. Only remainders 2 and 0 are appearing here.

Conclusion:

As there are only remainders 2 and 0 on division by 4, the even integers can be written as either 4q or 4q + 2 for anu whole number q.

Hence proved!!!

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