Question

1 (15 Points]. Prove the following statement: is divisible by 3 if only if it is a sum of three consecutive integers. be ddvisbk

Answer 1

Answer:

First. Let us prove that the sum of three consecutive integers is divisible by 3.

Three consecutive integers can be written as k, k+1, k+2. Then, if we denote their sum as n:

n = k+(k+1)+(k+2) = 3k+3 = 3(k+1).

So, n can be written as 3 times another integer, thus n is divisible by 3.

Second. Let us prove that any number divisible by 3 can be written as the sum of three consecutive integers.

Assume that n is divisible by 3. The above proof suggest that we write it as

n=3(k+1)=3k+3=k + k + k +1+2 = k + (k+1) + (k+2).

As k, k+1, k+2 are three consecutive integers, we have completed our goal.

Step-by-step explanation: