Question

Use Euclid's division lemma to show that any positive odd integer is of the form 6q+1,6q+3,6q+5 where q is a certain integer.

Answer 1

Heya,
Here is your answer,

Use Euclid's division lemma to show that any positive odd integer is of the form 6q+1,6q+3,6q+5 where q is a certain integer.

Answer:-
Let a be a positive odd integer
a=bq+r
b=6
a=6q+r, 0≤r<6. So,the possible values of r are 0,1,2,3,4,5

Set of positive odd integers are {1,3,5,7,9......}
put a=1,3,5,7,9......
a=bq+r
1=6(0)+1=6q+1 [r=1]
3=6(0)+3=6q+3 [r=3]
5=6(0)+5=6q+5 [r=5]
7=6(1)+1=6q+1 [r=1]
9=6(1)+3=6q+3 [r=3]

So,any positive integer is of the form 6q+1,6q+3,6q+5 where q is certain integer.

Hence showed.

Hope it helps

Answer 2

Let 'a' be any positive integer and b=6. Then, by Euclid's algorithm,a=6q+r, for  some integer q≥0, and r=0 or r=1 or r=2 or r=3 or r=4 or r=5, because 0≤r<6. So, a=6q or 6q+1 or 6q+2 or 6q+3 or 6q+4 or 6q+5.
If a is of the form 6q or 6q+2 or 6q+4, then a is an even integer. Also, a positive integer can be either even or odd. Therefore, any positive integer is of the form 6q+1,6q+3 and 6q+5.