Show that any positive integer is of the form 3q or, 3q +1 or 3q + 2 for some integers q
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Solution
let x be the integer x=3q x²=9q² x²=3(3q²) x²=3q [let 3q² be q] ============================================ x=3q+1 x²=(3q+1)² x²=9q²+6q+1 x²=3(3q²+2q)+1 x²=(3q+1) [let 3q²+2q be q] ============================================ x=3q+2 x²=(3q+2)² x²=9q²+12q+4 x²=3(3q²+4q+1)+1 x²=(3q+1) [3q²+4q+1 be q] ============================================= it proves q² is in the form of 3q and (3q+1) but not in the form of (3q+2)