MathematicsState true or false:The square of any positive integer is of the form 3m, 3m+1 but not of the form 3m+2.

State true or false:

The square of any positive integer is of the form 3m, 3m+1 but not of the form 3m+2.

  1. A
    True
  2. B
    False 

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    Solution:

    We have to find whether the square of any positive integer is of the form 3m, 3m+1 but not of the form 3m+2.
    Euclid's Division Lemma states that, if two positive integers “a” and “b”, then there exists unique integers “q” and “r” such that which satisfies the condition:
    a=bq+r where  0rb
    Consider ‘n’ as a positive integer for some integer q.
    From Euclid's Division Lemma for b=3,
    It can be of the form 3q, 3q+1 or 3q+2.
    Substitute n=3q and square on both sides,
    n2=3q2 n2=9q2 n2=3(3q2) Let 3q2 be ‘x’, then n=3x where x is some integer.
    Now substitute n = 3q+1 and square on both sides,
    n2=3q+12 n2=9q2+6q+1 n2=3(3q2+2q) Let (3q2+2q) be ‘x’, then n=3x+1 where x is some integer.
    Substitute n = 3q+2 in and square on both sides,
    n2=3q+22 n2=9q2+12q+4 n2=3(3q2+4q+1)+1 Let (3q2+4q+1) be ‘x’, n=3x+1 where x is some integer.
    Thus, the square of any positive integer is of form 3m or 3m+1 but not of form 3m+2.
    Therefore, it is true that the square of any positive integer is of the form 3m, 3m+1 but not of the form 3m+2.
    Hence, option 1 is correct.
     
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