MATH135 - Assignment5 Solutions

MATH135 - Assignment5 Solutions - MATH 135 Assignment#5...

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MATH 135 Fall 2007 Assignment #5 Due: Wednesday 24 October 2007, 8:20 a.m. Hand-In Problems 1. (a) Convert (84 808) 9 to base 10. Show your work. (b) Convert 37 390 to base 16, using A for ten, B for eleven, C for twelve, D for thirteen, E for fourteen, and F for fifteen. Show your work. 2. Add and multiply the numbers (3157) 8 and (445) 8 , expressing your answers in base 8. 3. (a) Determine the prime factorizations of 39 204 000 and 9 922 500. (b) Determine gcd(39 204 000 , 9 922 500) and lcm(39 204 000 , 9 922 500). (c) Determine the number of positive divisors of 9 922 500. (d) Determine the number of positive common divisors of 39 204 000 and 9 922 500 that are multiples of 10. 4. A positive integer n is called a perfect square if n = m 2 for some positive integer m . (a) If n is a perfect square and n has prime factorization n = p c 1 1 · · · p c k k , prove that c 1 ,c 2 ,...,c k are all even. (b) Suppose that a,b,c P with a,b,c > 1 and ab = c 2 . If gcd( a,b ) = 1, prove that a and b are both perfect squares. 5. Suppose a,b Z . (a) Prove that if a | b , then a 3 | b 3 . (b) Suppose a = p c 1 1 · · · p c k k and b = p d 1 1 · · · p d k k for some primes p 1 ,...,p k and some non-negative integers c 1 ,...,c k ,d 1 ,...,d k . Prove that if a 3 | b 3 , then a | b . 6. (a) Determine the number of zeroes with which the base 10 representation of 135! ends. Explain how you got your answer. (b) Determine the number of zeroes with which the base 16 representation of 135! ends. Explain how you got your answer. 7. Suppose that x is a digit in base 16 and y is a digit in base 7. If ( xx ) 16 = (2 y 0) 7 , determine x and y . 8. Every polyhedron obeys Euler’s Formula: V - E + F = 2, where V is the number of vertices, E is the number of edges and F is the number of faces. A polyhedron has 20 faces that are hexagons, p faces that are pentagons, and no additional faces. Three faces meet at each vertex (that is, three edges meet at each vertex). (a) In terms of p , determine the number of edges that the polyhedron has. (b) In terms of p , determine the number of vertices that the polyhedron has. (c) Determine the value of p . (This problem is not directly related to the course material, but is included to keep your problem solving skills sharp.)
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Recommended Problems 1. Text, page 51, #54 2. Text, page 51, #58 3. Text, page 52, #71 4. Text, page 54, #92 5. Text, page 55, #107 6. Text, page 55, #108 7. Suppose that p is a prime number with p > 3. (a) Prove that the remainder when p is divided by 4 is 1 or 3. (b) Prove that the remainder when p is divided by 6 is 1 or 5. 8. (a) Prove that if n 3, then n ! + 3 is not prime. (b) Prove that for every k P , k consecutive positive integers that are not prime can be found. (A good way to prove this is to explicitly show what these k integers could be.)
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MATH 135 Fall 2007 Assignment #5 Solutions Hand-In Problems 1. (a) (84 808) 9 = 8 × 9 4 + 4 × 9 3 + 8 × 9 2 + 0 × 9 + 8 = 52488 + 2916 + 648 + 0 + 8
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This note was uploaded on 09/21/2008 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.

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MATH135 - Assignment5 Solutions - MATH 135 Assignment#5...

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