Three consecutive integers have a sum of 72. Find the largest of them.

Step-by-step solution

1. 1

   Name the integers with one variable. Consecutive integers differ by $1$, so if the smallest is $x$ the three are

   $x,\quad x+1,\quad x+2$

   Using three separate letters would need two extra equations to say they are consecutive.

2. 2

   Translate the sum into an equation.

   $x+(x+1)+(x+2)=72$

3. 3

   Collect like terms and solve.

   $3x+3=72\;\Longrightarrow\;3x=69\;\Longrightarrow\;x=23$

4. 4

   Answer the exact question asked. The three integers are $23$, $24$ and $25$, and the problem asks for the largest:

   $x+2=25$

   $\boxed{25}$

5. 5

   Check and note a shortcut. $23+24+25=72$. ✓ Also, because the three numbers are evenly spaced, their mean equals the middle one: $72\div 3=24$, so the middle integer is $24$ and the largest is $24+1=25$ — a one-line route worth remembering.