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

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)#