# Use Euclid’s division lemma to show that the square of any positive integer is either of the form for some integer m. [Hint : Let be any positive integer then it is of the form . Now square each of these and show that they can be rewritten in the form .]

by euclid division lemma,we know that

If a and b are two positive integers, then,by euclid division lemma

a = bq + r, 0  r  b Let b = 3

Therefore, r = 0, 1, 2

Therefore, a = 3q or a = 3q + 1 or a = 3q + 2

If a = 3q:

If a = 3q + 1 :

If a = 3q + 2 :

Therefore, the square of any positive integer is either of the form 3m or 3m + 1.

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