Multiples of 3 and Consecutive Numbers

Numbers that are multiples of 3 are 3, 6, 9, 12, …..

Problem

Prove that numbers that are multiples of 3 can be expressed as sum of three consecutive numbers.

Solution (2 out of 5 marks)

3 = 0 + 1 + 2

6 = 1 + 2 +3

9 = 2 + 3 + 4

12 = 3 + 4 + 5

15 = 4 + 5 + 6

Comment: Generating examples means you understood the statement that multiples of 3 can always be expressed as sum of consecutive numbers. That’s why you get 2 marks. However, even if you listed hundreds of multiples of 3  and expressed each as sum of consecutive numbers you will still get 2 out of 5 marks. Showing that you understood the statement is not the same as proving it. Same with generating cases unless you can list all of them. I might consider giving an additional half-mark if you can list a million of them:-).

Solution (full mark)

Let n, n + 1, n + 2 be the three consecutive numbers, n is an integer

Sum is n + n + 1 + n + 2 = 3n + 3 = 3 (n + 1)

Since 3 is a factor in 3(n + 1), 3(n + 1) it is a multiple of 3.

Something to think about:

Can we always express a number as sum of at least two consecutive numbers?

Answer

No.

Proof?

The number 4.

Comment: One counterexample is enough to disprove a statement.

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