Question

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

Answer 1

Answer:

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