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how to express 3m or3m+r in Euclid division method by squaring of any 2 integer
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how to express 3m or3m+r in Euclid division method by squaring of ...
If a and b are two positive integers, then,
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 a2 = 9q2 = 3(3q2) = 3m where m = 3q2
If a = 3q + 1  a2 = 9q2 + 6q + 1 = 3(3q2 + 2q) + 1 = 3m + 1 where m = 3q2 + 2q
If a = 3q + 2 a2 = 9q2 + 12q + 4 = 3(3q2 + 4q + 1) + 1= 3m + 1, where m = 3q2 + 4q + 1
Therefore, the square of any positive integer is either of the form 3m or 3m + 1. 
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how to express 3m or3m+r in Euclid division method by squaring of any 2 integer Related: Facts that Matter - Real Numbers?
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