Use Euclid’s Division Algorithm to show that the squares of following numbers is either of form 3m or 3m + 1 for some integer m.
151
Euclid’s Division Algorithm: For any two positive integers a and b,
Assume:
Let a be any positive integer and b = 3.
Then a = 3q + r for some integer q ≥ 0
And r = 0, 1, 2 because 0 ≤ r < 3
Therefore, a = 3q or 3q + 1 or 3q + 2
Case 1: a = 3q
Case 2: a = 3q + 1
Case 3: a = 3q + 2
Thus, it can be said that the square of any positive integer is either of the form 3m or 3m + 1.
Hence, the square of 151 is either of the form 3m or 3m + 1.