## Show that every positive odd integar is of the from 6q+1 or 6q+3 or 6q+5 for some integer q

Question

Show that every positive odd integar is of the from 6q+1 or 6q+3 or 6q+5 for some integer q

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1 month 2021-08-20T05:18:21+00:00 1 Answer 0 views 0

We know that 6 is an even number….So it cant divide any odd number without remainders..

Now, the possible values when a positive integer N is divided by 6 are

6q            It can’t be this as it divides the number perfectly

6q + 1       There is a possibility as there is an odd remainder

6q + 2      It can’t be this as the remainder is even

6q + 3      There is a possibility as there is an odd remainder

6q + 4      It can’t be this as the remainder is even

6q + 5      There is a possibility as there is an odd remainder

Thus we get the three possibilities… Any odd positive integer can be represented in the form 6q + 1, 6q + 3 or 6q + 5

PS: What I mean be the  even remainders are that the whole equation can be divided by 2 and any number divisible by two is not an odd number