NOTE: It is also strongly recommended to review past homework problems and your notes from group work in class. The problems on this page do not represent every type of question that could occur on the test.

1. Find prime factorizations of the following numbers:
1. 28
   solution: 2²·7
2. 68
   solution: 2²·17
3. 323
   solution: 17·19
4. 108
   solution: 2²·3³
5. 109
   solution: it is prime. Check that numbers < 11 don't divide it.
    
2. Using the prime factorizations, find the following greatest common divisors and least common multiples:
1. GCD(28,68) and LCM(28,68)
   solution: GCD = 2² and LCM = 2²·7·17
2. GCD(28,108) and LCM(28,108)
   solution: GCD = 2² and LCM = 2²·3³·7
3. GCD(28,323)
   solution: GCD = 1
4. GCD(68,323)
   solution: GCD = 17
5. GCD(28,109)
   solution: GCD = 1
    
3. Use the Euclidean algorithm to find the following GCD's:
1. GCD(16,37)
   solution: 37 = 2·16 + 5, so that GCD(16,37) = GCD(16,5)
   16 = 3·5 + 1, so that GCD(16,5) = GCD(5,1) = 1.
   We conclude that GCD(16,37) = 1
2. GCD(28,109)
   solution: 109 = 3·28 + 25, so that GCD(28,109) = GCD(28,25)
   28 = 25 + 3, so that GCD(28,25) = GCD(25, 3)
   25 = 8·3 + 1, so that GCD(25, 3) = GCD(3,1) = 1.
   We conclude that GCD(28,109) = 1
3. GCD(1003,1999)
   solution: 1999 = 1003 + 996, so that GCD(1003,1999) = GCD(1003,996)
   1003 = 996 + 7 so that GCD(1003,996) = GCD(996, 7)
   996 = 142·7 + 3, so that GCD(996, 7) = GCD(7, 3)
   7 = 3·2 + 1, so that GCD(7,3) = GCD(3,1) = 1
   We conclude that GCD(1003,1999) = 1
    
4. In each case, find the next five numbers in the given base:
1. 11011_two
   solution: the next 5 numbers are 11100_two, 11101_two, 11110_two, 11111_two, 100000_two .
2. 2221_three
   solution: the next 5 numbers are 2222_three, 10000_three, 10001_three, t0002_three, 10010_three .
3. 2AD _sixteen
   solution: the next 5 numbers are 2AE_sixteen, 2AF_sixteen, 2B0_sixteen, 2B1_sixteen, 2B2_sixteen .
    
5. Convert to bases 2, 3, and 16:
1. 79
   solution: 79 = 64 + 8 + 4 + 2 + 1 = 2⁶ + 2³ + 2² + 2¹ + 2⁰ = 1001111_two
   
   79 = 2·27 + 2·9 + 2·3 + 1 = 2·3³ + 2·3² + 2·3¹ + 1·3⁰ = 2221_three
   
   79 = 4·16 + 15 = 4F_sixteen
   .
2. 257
   solution: 257 = 256 + 1 = 2⁸ + 2⁰ = 100000001_two
   
   257 = 243 + 9 + 3 + 2 = 3⁵ + 3² + 3¹ + 2·3⁰ = 100112_three
   
   257 = 256 + 1 = 16² + 1 = 101_sixteen .
    
6. Convert to base 10:
1. 100011_two
   solution: 100011_two = 2⁵ + 2¹ + 2⁰ = 32 + 2 + 1 = 35
2. 177_nine
   solution: 177_nine = 9² + 7·9 + 7 = 81 + 63 + 7 = 151
3. BBB _twelve
   solution: BBB _twelve = 11·12² + 11·12 + 11 = 1584 + 132 + 11 = 1727
   check: 12³ - 1 = 1728 - 1 = 1727
    
7. Do the indicated conversions:
1. 110111011_two to base eight
   solution: (110)(111)(011)_two = 673_eight
2. 21201_three to base nine
   solution: (02)(12)(01)_three = 251_nine .
    
8. Do the following calculations in the indicated bases. Show your work, including which numbers were carried. Check your answer by converting to base 10 and doing appropriate base 10 calculations:
1. Add in base eight: 6142_eight and 2457_eight eight
   
   solution:
    
2. Subtract in base eight: 651_eight from 6142_eight
   
   solution:
    
3. Multiply in base five: 423_five by 214 _five
   
   solution:

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For questions: roberts@math.umn.edu