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Question

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


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Solution

Euclid’s Division Algorithm: For any two positive integers a and b, a=bq+r,0r<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

a2=3q2=9q2=3×3q2=3m,wherem=3q2

Case 2: a = 3q + 1

a2=3q+12=9q2+6q+1=3×3q2+2q+1=3m+1,wherem=3q2+2q

Case 3: a = 3q + 2

a2=3q+22=9q2+12q+4=3×3q2+4q+1+1=3m+1,wherem=3q2+4q+1

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.


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