Question

Explain how you can tell whether an even number can be written a the sum of three consecutive whole numbers .

Answer 1

ASSUMPTIONS:

1) Let the even number be n.

2) Let the second of the 3 consecutive numbers be x. Therefore, we have the 3 consecutive numbers to be:

$(x-1),x,(x+1)$

PROOF:

We have that:

$n=(x-1)+x+(x+1)$

Solving the right-hand side:

$\begin{gathered} n=x-1+x+x+1 \\ n=3x \end{gathered}$

Since n is even, that means that x must be even.

We can get the value of x to by dividing both sides by 3 to get:

$x=\frac{n}{3}$

Since x is an even whole number, n must be divisible by 3.

CHECK:

Try n = 18:

$\begin{gathered} x=\frac{18}{3} \\ x=6 \end{gathered}$

Hence, the 3 numbers will be:

$\begin{gathered} (6-1),6,(6+1) \\ \Rightarrow5,6,7 \end{gathered}$

The sum is:

$5+6+7=18$

This proves the theory.

CONCLUSION:

For an even number to be able to be written as the sum of three consecutive whole numbers, it has to be DIVISIBLE BY 3.