# Real Numbers

#### Question 4:

Use
Euclid’s division lemma to show that the square of any positive
integer is either of form 3*m* or 3*m* + 1 for some integer
*m*.

[**Hint:
**Let *x* be any positive integer then it is of the form 3*q*,
3*q* + 1 or 3*q* *+ *2. Now square each of these and
show that they can be rewritten in the form 3*m* or 3*m + *1.]

#### Answer:

Let *a*
be any positive integer and *b* = 3.

Then *a*
= 3*q* + *r* for some integer *q* ≥ 0

And *r*
= 0, 1...

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