5.1 Number Theory 219 Example 6 illustrates this procedure. TO DETERMINE THE LEAST COMMON MULTIPLE OF TWO OR MORE NUMBERS 1. Determine the prime factorization of each number. 2. List each prime factor with the greatest exponent that appears in any of the prime factorizations. 3. Determine the product of the factors determined in Step 2. PROCEDURE MATHEMATICS TODAY Online Credit Card Number Safety Prime numbers play an essential role in protecting the credit card numbers of consumers who are making online purchases. The encryption or coding process, known as the RSA algorithm , relies on a key number, n, which is the product of two very large prime numbers, p and q. The prime numbers p and q are each over 150 digits long. The product, n, is very difficult to factor—even with the use of the world’s fastest computers. The number n is publicly available and is used by merchants to encrypt the credit card number after it is entered by the consumer. The prime numbers p and q are known only by the credit card company and are used to decrypt the credit card number. This same algorithm was used by the U.S. government for many years to protect key databases. Why This Is Important Encryption systems involve many areas of mathematics, including many that are discussed in this book, such as prime numbers, permutations, modular arithmetic, polynomials, and group theory. Encryption systems, relying on number theory, enable secure online transactions. Example 6 Using Prime Factorization to Determine the LCM Determine the LCM of 36 and 90. Solution 1. Determine the prime factors of each number. In Example 4, we determined that 36 2 3 and90 23 5 2 2 2 = ⋅ = ⋅ ⋅ 2. List each prime factor with the greatest exponent that appears in either of the prime factorizations: 2 , 3 , 5. 2 2 3. Determine the product of the factors determined in Step 2: 2 3 5 4 9 5 180 2 2 ⋅ ⋅ = ⋅ ⋅ = Thus, 180 is the LCM of 36 and 90. It is the smallest natural number that is divisible by both 36 and 90. 7 Now try Exercise 45 Example 7 Determining the LCM Determine the LCM of 315 and 450. Solution 1. Determine the prime factorization of each number. In Example 5, we determined that 315 3 57 and 450 23 5 2 2 2 = ⋅ ⋅ = ⋅ ⋅ 2. List each prime factor with the greatest exponent that appears in either of the prime factorizations: 2, 3 , 5 , 7. 2 2 3. Determine the product of the factors from Step (2): 23 5 7 29257 3150 2 2 ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ = Thus, 3150 is the LCM of 315 and 450. It is the smallest natural number that is divisible by both 315 and 450. 7 Now try Exercise 47 Oleksiy Mark/Shutterstock
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