Actually Applicable Application Problems and Brainteasers/Subtraction Trick for Common Factors

Overview
Rather than a real-world situation, this is a calculation trick that you can use to make other problems easier. Because subtraction is easier than division, it's Actually Applicable in the sense that it takes something you have to do and makes it easier.

General Method

 * 1) Subtract the smaller number from the bigger number. Now you have three numbers.
 * 2) Cross out the biggest of the three numbers. Now you have two numbers.
 * 3) As long as those two numbers are different, take them back to step 1.
 * 4) If those two numbers are actually the same, they are your Greatest Common Factor. All of its factors are other common factors of the two numbers you started with.

A note about why it works
Factors of a number are smaller numbers that you can skip-count by and get the original number.

A common factor of two numbers is a factor of both.

If you skip-count by the common factor and get to two particular other numbers, the difference between them must be some number of steps of that factor. In other words, the difference between those two numbers is another number with the same common factor.

Smaller numbers are easier to work with, so keep going as long as you can.

Simplifying Fractions
One step in simplifying fractions is finding a common factor of the numerator and denominator. Find a common factor of 4 and 18 to simplify 4/18.

Make Your Own Problem
Pick any two numbers you are interested in, or are using in a situation, and find their common factors.