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Question

Show that any positive odd integer is in the form of (6m + 1 )or (6m + 3) or (6 m + 5) Where m is some integer

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Solution

Dear student,

Let take a as any positive integer and b = 6.
we get a = 6m + r here r is remainder and value of m is more than or equal to 0 and r = 0, 1, 2, 3, 4, 5 because 0 <= r < b and the value of b is 6
So total possible forms will 6m+0 , 6m+1 , 6m+2,6m+3,6m+4,6m+5
6m+0 6 is divisible by 2 so it is a even number
6m+1 6 is divisible by 2 but 1 is not divisible by 2 so it is a odd number
6m+2 6 is divisible by 2 and 2 is also divisible by 2 so it is a even number
6m+3 6 is divisible by 2 but 3 is not divisible by 2 so it is a odd number
6m+4 6 is divisible by 2 and 4 is also divisible by 2 it is a even number
6m+5 6 is divisible by 2 but 5 is not divisible by 2 so it is a odd number
So odd numbers will in form of 6m + 1, or 6m + 3, or 6m+ 5

Thank you.

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