The sports day at a school of eighty-four pupils begins with a parade, and the parade begins with an argument in the staff room.
There are 48 boys and 36 girls. The head teacher wants them in rows, boys with boys and girls with girls, and the same number in every row so that the lines look straight from the stand.
Rows of ten are impossible: ten does not divide either number. Rows of eight fit the boys but leave four girls standing on their own. Rows of six work for both — eight rows of boys and six of girls — and for two years that is what the school did.
Then a mathematics teacher pointed out that six is not the only number that divides both numbers exactly. Two does, three does, four does, six does — and so does twelve. Twelve is the largest that works: it gives four rows of boys and three of girls, seven rows in all, and a parade that takes half the time to form up.
Anything larger fails. Sixteen divides 48 but not 36; eighteen divides 36 but not 48. Twelve is the point where both numbers agree, and there is nothing above it.
The problem the school solved that morning has a name in every algebra book: the greatest common factor. It is the first thing you take out of an expression, and the reason is the same in both places — take the largest piece that fits everything, and what is left is as simple as it can be.