5. (Modular Inverses) a) Just as we have inverses under the usual addition and multiplication operations,...

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5. (Modular Inverses) a) Just as we have inverses under the usual addition and multiplication operations, we can also define inverses in modular arithmetic too. For example, the inverse of a in addition modulo m is the number b (if it exists) such that a + b = 0 mod m. Carefully define the multiplicative inverse of a number modulo m. b) Draw a picture that demonstrates how you envision modular arithmetic modulo 8 and modulo 10. Then use two copies of each picture to pair up the additive inverses and the multiplicative inverses modulo each base by drawing a line or other suitable connector between them. c) Prove that a has an inverse mod m when alm. d) What is the converse of (c)? Do you think that it is true or false based on your work in (c)? (You do not need to prove your answer) 5. (Modular Inverses) a) Just as we have inverses under the usual addition and multiplication operations, we can also define inverses in modular arithmetic too. For example, the inverse of a in addition modulo m is the number b (if it exists) such that a + b = 0 mod m. Carefully define the multiplicative inverse of a number modulo m. b) Draw a picture that demonstrates how you envision modular arithmetic modulo 8 and modulo 10. Then use two copies of each picture to pair up the additive inverses and the multiplicative inverses modulo each base by drawing a line or other suitable connector between them. c) Prove that a has an inverse mod m when alm. d) What is the converse of (c)? Do you think that it is true or false based on your work in (c)? (You do not need to prove your answer)

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a The multiplicative inverse of a number a modulo m is the number b if it exists such that ab 1 mod ... View the full answer