Question

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

Answer 1

Let take a as any positive integer and b = 6.


Then using Euclid’s algorithm we get a = 6q + rhere 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.