Show that every positive odd integer in of the form (6m+1) or (6m+3) o

1.	Show that every positive odd integer in of the form (6m+1) or (6m+3) or (6m+5).
Answer» Let n be a given positive odd integer.On dividing n by 6, let m be the quotient and r be the remainder.Then, by Euclid\'s division lemma, we have{tex}n = 6m + r{/tex}, where {tex}0 \\leq r < 6{/tex}{tex}\\Rightarrow{/tex}\xa0{tex}n = 6m + r{/tex}, where {tex}r = 0, 1, 2, 3, 4, 5{/tex}{tex}\\Rightarrow{/tex}\xa0{tex}n = 6m\\ or\\ (6m + 1){/tex}{tex}or\\ (6m + 2)\\ or\\ (6m + 3)\\ or\\ (6m + 4)\\ or\\ (6m + 5).{/tex}But, {tex}n = 6m, (6m + 2), (6m + 4){/tex} give even values of n.Thus, when n is odd, it is of the form {tex}(6m + 1)\\ or\\ (6m + 3)\\ or\\ (6m + 5){/tex} for some integer m.

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