# Properties of Numbers    Greatest Common Factors The Greatest Common Factor (G.C.F.) of two numbers is the largest number that is a divisor of both. It is sometimes called the Greatest Common Divisor. It can be used to simplify (or reduce) fractions. Don't let the "greatest" in the name fool you - the GCF is no larger than the smallest of the numbers. Hints: GREATEST is biggest or largest. COMMON is something shared or in common FACTORS are the parts of multiplication facts. EXAMPLE: Find the Greatest Common Factor (G.C.F.) of 6 and 10. 6 = 2 * 3 You can divide 6 by 2 or by 3 6 = 1 * 6 You can divide 6 by 1 or by 6 Therefore 1, 2, 3, and 6 are all factors of six. 10 = 2 * 5 You can divide 10 by 2 or by 5 10 = 1 * 10 You can divide 10 by 1 or by 10 Therefore 1, 2, 5, and 10 are all factors of ten. Both 6 and 10 can be divided by 1 and by 2; 2 is greater than 1, so 2 is the Greatest Common Factor (G.C.F.) of 6 and 10. You can also use the prime factorization method to find the Greatest Common Factor: EXAMPLE: Find the Greatest Common Factor (G.C.F.) of 36 and 54. 36  = 2*2*3*3 54  = 2*3*3*3 Both have 2*2*3 in common: G.C.F = 2*3*3 = 18 Usually you can find the Greatest Common Factor fairly easily by experimenting with possible divisors: Start with the smaller number; it is the largest divisor of itself. Is it a divisor of the larger number? If so, you have the G.C.F.; if not: What is the next-largest divisor of the smaller number; is IT a divisor of the other number? Continue until you find a number that will divide into BOTH. Sometimes only the number '1' will work as a common divisor; for example: 21 and 16 have no common factor other than 1. EXAMPLE: Find the Greatest Common Factor (G.C.F.) of 10 and 30. Will 10 go into 30? Yes: G.C.F. = 10 Find the Greatest Common Factor (G.C.F.) of 24 and 18. Will 18 go into 24? No. Next pair: 2 * 9 = 18 Will 9 go into 24? No. Next pair: 3 * 6 = 18 Will 6 go into 24? Yes: G.C.F. = 6     