0
You visited us 0 times! Enjoying our articles? Unlock Full Access!
Question

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

Solution
Verified by Toppr

Using Euclid division algorithm, we know that a=bq+r, 0rb ----(1)

Let a be any positive integer and b=6.

Then, by Euclid’s algorithm, a=6q+r for some integer q0, and r=0,1,2,3,4,5 ,or 0r<6.

Therefore, a=6qor6q+1or6q+2or6q+3or6q+4or6q+5

6q+0:6 is divisible by 2, so it is an even number.

6q+1:6 is divisible by 2, but 1 is not divisible by 2 so it is an odd number.

6q+2:6 is divisible by 2, and 2 is divisible by 2 so it is an even number.

6q+3:6 is divisible by 2, but 3 is not divisible by 2 so it is an odd number.

6q+4:6 is divisible by 2, and 4 is divisible by 2 so it is an even number.

6q+5:6 is divisible by 2, but 5 is not divisible by 2 so it is an odd number.

And therefore, any odd integer can be expressed in the form 6q+1or6q+3or6q+5

Was this answer helpful?
28
Similar Questions
Q1
Show that any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.
View Solution
Q2

Show that any positive odd integer of the form 6q+ 1 or 6Q + 3 or 6Q + 5 were q in some integer

View Solution
Q3
Use division algorithm to show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer.
View Solution