. 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

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

Applying Euclid’s  algorithm

a = 6q+r               

Here, r= remainder= 0,1,2,3,4,5 and q>0

So, total possible forms are  were

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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