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Show that every positive odd integer …

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Show that every positive odd integer in of the form (6m+1) or (6m+3) or (6m+5).

Posted by Pramod Kumar 6 years, 6 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 6 months ago

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} {tex}n = 6m + r{/tex}, where {tex}r = 0, 1, 2, 3, 4, 5{/tex}
{tex}\Rightarrow{/tex} {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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