Question

show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer.

Answer 1

Let a be any positive odd integer and b be 6

By using the euclids lemma we get

When r= 0, a= 6q. (even)

r=1, a=6q+1. (odd)

r=2, a=6q+2. (even)

r=3, a=6q+3. (odd)

r=4, a=6q+4. (even)

r=5, a=6q+5. (odd)

As 6q+1, 6q+3 and 6q+5 are odd

Therefore for any positive odd integer a we can write in the form 6q+1 , 6q+3 and 6q+5

Answer 2

Let be assume x= 6m,6m+1 , 6m+2, 6m + 3,6m+4, 6m+5.
In case 1= (6m)
3(2m )
3q, where as q=2m
In case 2 = (6m+1)
3(2m)+1
3q+1, where as q=2m
In case 3= (6m+2)
3(2m)+2
3q+2, where as q= 2m
In case 4 = (6m + 3)
3(2m+1)
3q, where as q= (2m+1)
In case 5=(6m+4)
3(2m)+4
3q+4 , where as q= 2m
In case 6 = (6m+5)
3(2m)+5
3q+5 ,where as q = 2m
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