# GRE Arithmetic - Factorization and Least Common Multiples

Factorization and least common multiple questions appear frequently on the GRE exam. To **factorize** any number means to find the prime numbers that are factors of the original number. Factorization is simple if the original number is small. For example, 21 is the product of the prime numbers 3 and 7.

If you want to factor a large number, choose 2 factors and check to see if they are prime numbers. If neither or only one of them is a prime, factor the non-prime number(s). Continue this factorization process until you are left with only prime numbers.

As an example, factorization of the number 240 is done as follows:

Step 0: | Original number | 240 |

Step 1: | Divide 240 by 10 | 240 = 24 x 10 |

Step 2: | Divide 24 by 6 | 240 = 4 x 6 x 10 |

Step 3: | Divide 4 by 2 | 240 = 2 x 2 x 6 x 10 |

Step 4: | Divide 6 by 3 | 240 = 2 x 2 x 3 x 2 x 10 |

Step 5: | Divide the 10 by 5 | 240 = 2 x 2 x 3 x 2 x 5 x 2 |

From the above steps, you see that 240 is factorized into 240 = 2^{4} x 3 x 5, and so the prime factors of 240 are 2, 3 and 5. Basically, notice that you keep on dividing numbers as long as there is a number left to divide. Once you are left with only prime numbers, then you know that you've completed the factorization.

The **Least Common Multiple (LCM)** of two or more integers is the smallest integer that is a multiple of each of them. For example, the LCM of 5 and 3 is 15, and the LCM of 22 and 3 is 66.

The **Greatest Common Factor (GCF)** of two or more integers is the greatest integer that is a factor of each of the original numbers. For example, the GCF of 12 and 9 is 3, because the prime factors of 12 are 3 and 2, and the factor of 9 is 3, and so 3 is the largest common factor.