MATH135 - Assignment6 Solutions

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 6 | 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 6 = 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.)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/21/2008 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.

Page1 / 7

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online