MATH135 - Assignment6 Solutions

MATH135 - Assignment6 Solutions - MATH 135 Assignment#6...

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MATH 135 Fall 2007 Assignment #6 Due: Wednesday 31 October 2007, 8:20 a.m. Hand-In Problems 1. In each part, explain how you got your answer. (a) What is the remainder when 14 585 is divided by 3? (b) Is 8 24 + 13 12 divisible by 7? (c) What is the last digit in the base 6 representation of 8 24 ? (d) Determine the remainder when 42 2007 + 2007 10 is divided by 17. 2. Suppose that p is a prime number. (a) Prove that if x y (mod p ), then x n y n (mod p ) for every n P by induction on n . (This proof is not difficult mathematically. We will be looking for a very carefully written proof here.) (b) Using the definition of congruence, prove that if x 2 y 2 (mod p ), then x ≡ ± y (mod p ). (c) Determine the number of integers a with 0 a < 4013 with the property that there exists an integer m with m 2 a (mod 4013). (Note that 4013 is a prime number.) (d) Disprove the statement “If x 4 y 4 (mod p ), then x ≡ ± y (mod p )”. 3. (a) Prove that 5 n 7 + 7 n 5 + 23 n 0 (mod 7) for all n P . (b) Prove that 35 | 5 n 7 + 7 n 5 + 23 n for all n P 4. Let p be a prime number. Fermat’s Little Theorem tells us that if p | a , then a p - 1 1. But p - 1 might not be the smallest positive integer k for which a k 1 (mod p ). (a) Find a prime p > 5, a positive integer b > 1 that is not divisible by p and a positive integer k < p - 1 for which b k 1 (mod p ). (b) Suppose that p is a prime number, a is a positive integer not divisible by p , and s is the smallest positive integer for which a s 1 (mod p ). Prove that s | p - 1. (Hint: Start by dividing p - 1 by s , giving quotient q and remainder r .) 5. For each of the following congruences, determine if there exists a positive integer k that makes the congruence true. If so, determine the smallest integer k that works and justify why the integer that you’ve found is indeed the smallest such integer. If not, explain why not. (a) 2 k 1 (mod 18) (b) 8 k 1 (mod 17) 6. Suppose that A ( a, a 2 ) and B ( b, b 2 ) are points on the parabola y = x 2 , with a = b . (a) Write down the coordinates of the midpoint M of AB . (b) The line is tangent to the parabola at P so that is parallel to AB . Determine the coordinates of point P . (c) Determine the equations of the tangent lines to the parabola at A and B , and the coordi- nates of their point of intersection, Q . (d) Prove that M , P and Q are collinear (that is, lie on the same straight line). (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 82, #4 2. Text, page 83, #31 3. Text, page 85, #74 4. Text, page 85, #76 5. Suppose that a = ( r n r n - 1 · · · r 2 r 1 r 0 ) 10 . (a) Prove that a is divisible by 8 if and only if ( r 2 r 1 r 0 ) 10 is divisible by 8.
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