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.