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 take a as any positive integer and b = 6.
Then using Euclid’s algorithm we get a = 6q +
r here r is remainder and value of q is more than or
equal to 0 and r = 0, 1, 2, 3, 4, 5 because 0 <= r <
b and the value of b is 6
So total possible forms will 6q+0 , 6q+1 , 6q+2,6q
+3,6q+4,6q+5
6q+0 6 is divisible by 2 so it is a even number
6q+1 6 is divisible by 2 but 1 is not divisible by 2
so it is a odd number
6q+2 6 is divisible by 2 and 2 is also divisible by 2
so it is a even number
6q+3 6 is divisible by 2 but 3 is not divisible by 2
so it is a odd number
6q+4 6 is divisible by 2 and 4 is also divisible by 2
it is a even number
6q+5 6 is divisible by 2 but 5 is not divisible by 2
so it is a odd number
So odd numbers will in form of 6q + 1, or 6q + 3,
or 6q + 5