Question

The sum of three consecutive integers is 1,623. What are the numbers?

Answer 1

The three consecutive integers are `540, 541, 542`.

Explanation:

Three consecutive integers are three numbers in a row. For example, 4, 5 and 6 are three consecutive integers. If you start with the first number, you get the second number by adding 1 to the first number (4+1=5). You get the third number by adding 2 to the first number (4+2=6).

Let's call the first number (integer) `color(blue)x`.

Find the second number by adding 1 to the first. So the
2nd consecutive integer is `color(red)(x+1)`

Find the 3rd number by adding 2 to the first. The 3rd consecutive integer is `color(limegreen)(x+2)`.

The problem also states that the sum of the three consecutive integers is `1,623`. The word "sum" means the answer to an addition problem. So, we add the three numbers and set them equal to `color(magenta)(1,623)`.

`color(blue)xcolor(white)(aa)+color(red)(x+1)color(white)(aa)+color(limegreen)(x+2)=color(magenta)(1623)`

Combine like terms. First, add the three x's.

`3x+ 1 +2=1623`

Next, add the 1 and the 2.

`3x+3=1623`
`color(white)a`
`color(white)(aa)-3color(white)(aaa)-3color(white)(aaaa)`Subtract 3 from both sides.

`3x=1620`

Divide both side by 3.

`(3x)/3=1620/3`

`color(blue)x=540`

The first consecutive integer is `color(blue)(540)`.

Find the 2nd number by adding 1 to the first.
The 2nd consecutive integer is `540+1=color(red)(541)`

Find the 3rd number by adding 2 to the first.
The 3rd consecutive integer is `540+2=color(limegreen)(542)`

These three number "in a row" are three consecutive integers. Their sum is 1623. Let's check:

`color(blue)(540)+color(red)(541)+color(limegreen)(542)=color(magenta)(1623)`