Prove or disprove for any integer x one of the integers x x

Prove or disprove: for any integer x, one of the integers x, x + 2, x + 4 is divisible by 3. Prove or disprove: for any integer x, one of the integers x, x + 2, x + 8 is divisible by 3. Prove or disprove: for any integer x, one of the integers x, x + 5, x + 7 is divisible by 3. Formulate a (necessary and sufficient) condition on a, b such that the following statement is true: for any integer x, one of the integers x, x + a, x + b is divisible by 3.

Solution

(a)

x+4=x+1 mod 3

So modulo 3

three numbers are equivalent to ,x,x+1,x+2 ie three consecutive integers and hence one of them must be divisible by 3

(b)

x+8 =x+6+2=x+2 mod 3

Hence not true. For example. Let,

x=2,

2,2+2,2+8 gives

2,4,10 none of which are divisible by 3

(c)

x+5=x+3+2=x+2 mod 3

x+7=x+6+1=x+1 mod 3

Hence numbers are equivalent to ,x,x+1,x+2 modulo 3

Hence one of them must be divisible by 3

d)

As we have seen before the necessary and sufficient condition is that the three numbers must be equivlent to :x,x+1,x+2 mod 3

So two cases:

1. a=1 mod 3 and b=2 mod 3

or

2. a=2 mod 3 and b=1 mod 3