Use Euclid's division lemma to show that the square of any positive integer is either of the form `3m`or `3m+1`for some integer `m`.[Hint: Let x be any positive integer then it is of the form `3q`, `3q+1`or `3q+2`Now square each of these and show that they can be rewritten in the form `3m` or `3m + 1`].

Explanation

`a = 3q + r`

(Euclid's) `3 lt r le 0 or 1 or 2 `

`a = 3q or 3q+1 or 3q+2`

`a^2 = (3q)^2 or (3q + 1)^2 or (3q+2)^2`

`= 9q^2 or 9q^2+6q+1 or 9q^2+12q+4`

`= 3(3q^2)or 3(3q^2 + 2q)+1 or 3(3q^2+ 4q+1)+1`

let `(3q^2) & (3q^2 + 2q) & (3q^2+ 4q+1)` be m , n and p respectively

`a^2 = 3m or 3n+1 or 3p+1`

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