Question

Using Euclid’s division Lemma for any natural number and 4, which of the following forms can a positive integer be represented as, where m is an integer?

• A. 4m
• B. 4m + 1
• C. 4m + 2
• D. All of the above.

Answer 1

The correct option is D All of the above.

Let n be an arbitrary positive integer

On dividing n by 4, let m be the quotient and r be the remainder.

Then, by Euclid’s Division Lemma, we have

n = 4m + r , where $0\le r<4$
n = 4m , n = 4m + 1 , n = 4m + 2 and n = 4m + 3 for some integer m.

Thus, all the options are true.