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Answer:
Let a be a positive integer
Then, by Euclid division lemma
a = bq +r
Therefore, a = 3q +r - (1)
0 <r<b
0<r<3
r = 0,1,2
For, r =0
a = 3q
Squaring on both sides
a2 = 9 q 2
a2 = 3 × 3q 2
Since, a2 = 3m
Therefore, a2 = 3 m
For, r=1
a = 3 q +1
Squaring on both sides
a2 = (3q +1)2
a2 =9q2 + 6q +1
= 3 (3q2 + 2q) +1
m = 3q 2 + 2 q
a 2 = 3m +1
For, r = 2
a = 3 q + 2
Squaring on both sides
a 2 = ( 3 q + 2) 2
a2 = 9q2 + 12 q + 4
= 9 q 2 + 12 q + 3 + 1
= 3 ( 3 q 2 + 4 q + 1) + 1
m= 3q 2 + 4q + 1
a2 = 3m + 2
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