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Number Theory II: Worksheet |Solutions

Math 347, Summer 2019 Number Theory II: Worksheet Solutions Hildebrand Number Theory II: Worksheet Solutions The following problems illustrate some of the main applications of congruences. Some of the problems will be worked out in class, others will be part of the homework assignments. 1. Divisibility properties of large numbers: (a) Show that 3 divides 4n 1 for all n N. Solution: The claim is equivalent to 4n 1 0 mod 3 for all n N. Using the properties of congruences, this can be proved as follows: 4 1 mod 3, 4n 1n = 1 mod 3, 4n 1 0 mod 3. (b) Find the remainder of 31001 when divided by 5. Solution: 34 = 81 1 mod 5, 31001 (34 )250 3 1250 3 = 3 mod 5. 1001. (c) Find the reminder of 347 when divided by 3.

Next, we use the division algorithm to represent the given exponent 347 as a multiple of this (small) exponent we have found plus a remainder: 347 = 4 86 + 3: Finally, we use the properties of congruences and the fact that 34 1 mod 10 to nd the congruence sought: 3347 = 34 86+3 = (34)86 33 186 27 7 mod 10: Hence the last digit of 3347 in base ...

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