Question

5. Show that one and only one out of n, n + 2orn +4 is divisible by 3, where n is any positive integer.​

Answer 1

Let a nd b are any two positive integers there exist a unique integer 'q' and 'r' satisfying the equation

a=b q+r, where 0 less than or equal to 'r'less than b

When b=3, a=3q +r , where 0 less than or equal to r less than 3

(possible remainder 0,1,2)

When r= 0 ,a= 3q+0=3q

Case 1 ,

• n=3q ____ divisible by 3 ____ 1 st equation
• n+2=3q +2 _____ not divisible
• n+4=3q +4 _____ not divisible

when r =1

a= 3q +1

Case 2,

• n=3q+1 ______ not divisible
• n+2 = 3q +1+2 = 3q + 3 ___divisible by 3 __equation 2

• n+4 = 3q +1 +4 = 3q +5 ______ not divisible

When r = 2,

a = 3q + 2

Case 3,

• n=3q + 2 _____ not divisible
• n + 2 = 3q +2 + 2 _____ not divisible
• n + 4 = 3q + 2 + 4 = 3q + 6 ____ divisible by 3 ____ equation 3..

• So, one out of n , n+2 , n+4 is divisible by 3