Exercise 36 Page 183

We want to write the prime factorization of 45. The prime factorization of a composite number is the number written as a product of its prime factors. We will use factor pairs and a factor tree to find the prime factorization of 45. The factor tree is complete when only prime factors appear in the product.

The prime factorization of 45 is 3* 3* 5, or 3^2* 5.

The greatest common factor (GCF) is the greatest number that is a factor of two or more numbers. The first step for finding a GCF is to break down each number into its prime factors. We will write down all of the prime factors that are common between the given numbers.

Given number	Prime Factorization	Common Prime Factors
36	2 * 2 * 3 * 3	3, 3
45	3 * 3 * 5

The common prime factors are 3 and 3. The next step is to calculate the product of these factors. 3* 3 = 9 The GCF for 36 and 45 is 9.

The least common multiple (LCM) is the least multiple shared by two or more numbers, not including zero. The first step to find the LCM of two or more numbers is to write the prime factorization of each number.

Given Number	Prime Factorization
36	2 * 2 * 3 * 3
45	3 * 3 * 5

Then, we list the greatest number of times each factor appears in either prime factorization and multiply these factors. 2 * 2 * 3 * 3 * 5 = 180 The LCM for 36 and 45 is 180.