9.3 Modular Arithmetic 563 we subtract the modulus, 5, from 7, we obtain 2. Thus, 2 and 7 are in the same modular class. Therefore, ? 2. = − ≡ − ≡ − ≡ ? 4 3 (mod 5) 7 4 3 (mod 5) 2 4 3 (mod 5) Notice in the last equivalence that if we add 5 to 2, we get 7 4 3 (mod 5), − ≡ which is a true statement. c) 5 ? 7 (mod 8) − ≡ In mod 8, adding 8, or a multiple of 8, to a number results in a sum that is in the same modulo class. Thus, we can add 8 to 5 so that the statement becomes (8 5) ? 7 (mod 8) 13 ? 7 (mod 8) + − ≡ − ≡ We can see that 13 6 7. − = Therefore, ? 6 = and 5 6 7 (mod 8). − ≡ 7 Now try Exercise 47 Example 4 Using Modulo Classes in Multiplication Determine all positive number replacements, less than the modulus, for the question mark that make the statement true. a) 2 ? 3 (mod 5) ⋅ ≡ b) 3 ? 0 (mod 6) ⋅ ≡ c) 3 ? 2 (mod 6) ⋅ ≡ Solution a) One method of determining the solution is to replace the question mark with the numbers 0, 1, 2, 3, and 4 and then determine the equivalent modulo class of the product. We use the numbers 0 4 − because we are working in modulo 5. ⋅ ≡ ⋅ ≡ ⋅ ≡ ⋅ ≡ ⋅ ≡ ⋅ ≡ 2 ? 3 (mod 5) 2 0 0 (mod 5) 2 1 2 (mod 5) 2 2 4 (mod 5) 2 3 1 (mod 5) 2 4 3 (mod 5) Therefore, the question mark can be replaced with 4, since 2 4 3 (mod 5). ⋅ ≡ b) Since we are working in modulo 6, replace the question mark with the numbers 0 5 − and follow the procedure used in part (a). ⋅ ≡ ⋅ ≡ ⋅ ≡ ⋅ ≡ ⋅ ≡ ⋅ ≡ ⋅ ≡ 3 ? 0 (mod6) 3 0 0 (mod 6) 3 1 3 (mod6) 3 2 0 (mod 6) 3 3 3 (mod6) 3 4 0 (mod 6) 3 5 3 (mod 6) Therefore, replacing the question mark with 0, 2, or 4 results in true statements. The answers are 0, 2, and 4. c) 3 ? 2 (mod 6) ⋅ ≡ Examining the products in part (b) shows there are no values that satisfy the statement. The answer is “no solution.” 7 Now try Exercise 49 Profile in Mathematics Amalie Emmy Noether (1882–1935) Professor Amalie Emmy Noether was described by Albert Einstein as the “most significant creative mathematical genius thus far produced since the higher education of women began.” Because she was a woman, she was never to be granted a regular professorship in her native Germany. In 1933, with the rise of the Nazis to power, she and many other mathematicians and scientists emigrated to the United States. That brought her to Bryn Mawr College, at the age of 51, to her first full-time faculty appointment. She died just 2 years later. Her special interest lay in abstract algebra, which includes such topics as groups, rings, and fields. She developed a number of theories that helped draw together many important mathematical concepts. Her innovations in higher algebra gained her recognition as one of the most creative abstract algebraists of modern times.
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