Multiples

　　A multiple is an integer that can be evenly divided by another integer. If c /d is an integer, then c is a multiple of d. The numbers 45, 27, and 18, for example, are all multiples of 9. Alternatively, you could define a multiple as an integer with at least one factor. All that really matters is that you understand the concept of multiples, and this is best done with a simple example.　　What are some multiples of 4? 12, 20, and 96 are all multiples of 4. 　　How do we know these numbers are multiples of 4?　　Also, note that any integer, n, is a multiple of 1 and n, because 1　n = n.　　Least Common Multiple　　The least common multiple (LCM) of two integers is the smallest multiple that the two numbers have in common. The LCM of two numbers is, like the GCF, useful when manipulating fractions:　　For example, what is the least common multiple of 4 and 6? We must first find their prime factorizations.　　Their LCM is the smallest prime factorization that contains every prime number in each of the two original prime factorizations. For the numbers 4 and 6, this is 2　2　3 = 12. It is the smallest prime factorization that includes 2　2　3. Thus, 12 is the LCM of 4 and 6.　　Let’s try a harder example. What is the LCM of 14 and 38? Again, we start by finding the prime factorizations of both numbers: 　　Therefore, their LCM is 2　7　19 = 266.　　For some quick practice, find the LCM of the following pairs of integers:12 and 32 15 and 26 34 and 40 3 and 17 18 and 16　　Compare your answers to the solutions:12 = 23　3. 32 = 25. The LCM is 25　3 = 96. 15 = 3　5. 26 = 2　13. The LCM is 2　3　5　13 = 390. 34 = 2　17. 40 = 23　5. The LCM is 23　5　17 = 680. 3 = 1　3. 17 = 1　17. The LCM is 3　17 = 51. 18 = 2　32. 16 = 24. The LCM is 24　32 = 144.