show the every positive odd integer is of the from 6q+1,or 6q+3 or 61+5,where q is any integer
Answers
let a be a positive odd no. and b be 6. so it can be represented in the equation a=bq+r where 0≤r<6.
case 1: r=0
a=6q.
as 6 is an even no. so the product of 6q will also be an even no. so no positive odd no. can be of this form
case 2: r=1
a= 6q+1
as 6q is even, so the addition of 1 to the no. will give an odd no. so any positive odd no. can be of this form
case 3: r=2
a=6q+2
similarly this no. is even and a cant be of this form
case 4: r=3
a=6q+3
a can be of this form as 6q+3 is odd
case 5: r=4
a=6q+4
the no. 6q+4 is even so no. positive integer can be of this form
case 6: r=5
a=6q+5
a can be of this form as 6q+5 is an odd no. only.
case 7: r=6
a=6q+6
a cant be of this form as 6q+6 is again even.
therefore, any positive odd no. can only be in the form 6q+1, 6q+3 or 6q+5.
hope it helped