What three consecutive numbers add up to 417?

The secret to finding this answer is simply to define your terms using a single variable. To put it simply, how can you represent three consecutive numbers using only the variable x?

Let's make x the middle of the three numbers. The other two numbers in this consecutive set will therefore be x – 1 (which is the same as x + –1) and (x + 1). Now you just need to add these up and set them equal to 417:

(x + –1) + x + (x + 1) = 417

Because of the associative property of addition, you can eliminate the parentheses and regroup the numbers:

x + x + x + 1 + –1 = 417
3x = 417
x = 139

So the middle number is 139, which makes your three consecutive numbers 138, 139, and 140. To be safe, you should always test your answer:

138 + 139 + 140 = ???

This equation works out so cleanly because 417 is a multiple of 3, and multiples of 3 have two characteristics that aren't often mentioned:

  • The sum of any three consecutive numbers is always a multiple of 3.
  • Every multiple of three is the sum of three consecutive numbers.

This can give you a shortcut if you ever come across a problem like this again. If the final number isn't a multiple of 3, you can't add three consecutive numbers together to achieve it. If it is a multiple of 3, just divide the number by 3 and add the numbers on either side of that answer.

Incidentally, this doesn't work only for 3, but for all odd numbers. For example, the sum of every five consecutive numbers is a multiple of 5; the sum of seven consecutive numbers is a multiple of 7; and so on.