. Show that any positive odd integer is of the form


6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

 

Best Answer

Solution:

 

By taking,’ a’ as  any positive integer and b = 6.

Applying Euclid’s  algorithm

a = 6q+r               

Here, r = remainder = 0,1,2,34,5 and q>0

So, total possible forms are  6q+0 ,6+ 1,6q+2 or 6+ 3, 6q+4 or 6+ 5

6q+0 , (6 is divisible by 2 , its an even number)

6q+1, ( 6 is divisible by 2 but 1 is not divisible by 2 , its an odd number)

6q+2, (6 and 2 both are divisible by 2 , its an even number)

6q+3, (6 is divisible by 2 but 3 is not divisible by 2, its an odd number)

6q+4, ( 6 and 4 both are divisible by 2, its an even number)

6q+5 , (6 is divisible by 2 but 5 is not divisible by 2 , its an odd number)

Therefore,

So, odd numbers will be  in the form 6q+1, or 6q+3, or 6q+5

 

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