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# mod2 - AwesomeMath 2007 Track 4 Modulo Arithmetic Week 2...

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AwesomeMath 2007 Track 4 — Modulo Arithmetic Week 2 Lecture 6 : Divisibility and the Euclidean Algorithm Yufei Zhao July 24, 2007 1. If a and b are relatively prime integers, show that ab and a + b are also relatively prime. 2. (a) If 2 n + 1 is prime for some integer n , show that n is a power of 2. (b) If 2 n - 1 is prime for some integer n , show that n is a prime. 3. Show that the fraction 12 n + 1 30 n + 2 is irreducible for all positive integers n . 4. Let x, a, b be positive integers, show that gcd( x a - 1 , x b - 1) = x gcd( a,b ) - 1. 5. (a) Let p be a prime number. Determine the greatest power of p that divides n !, where n is a positive integer. (b) Let m and n be positive integers. Show that ( m + n )! m ! n ! is an integer (without referring to binomial coefficients). 6. (USAMO 1972) Show that gcd( a, b, c ) 2 gcd( a, b ) gcd( b, c ) gcd( c, a ) = lcm( a, b, c ) 2 lcm( a, b ) lcm( b, c ) lcm( c, a ) . 7. (a) Show that if a and b are relatively prime integers, then gcd( a + b, a 2 - ab + b 2 ) = 1 or 3. (b) Show that if a and b are relatively prime integers, and p is an odd prime, then gcd a + b, a p + b p a + b = 1 or p. 8. Let n be a positive integer. (a) Find n consecutive composite numbers. (b) Find n consecutive positive integers, none of which is a power of a prime. 9. Let n > 1 be a positive integer. Show that 1 + 1 2 + 1 3 + · · · + 1 n is not an integer. (Try not to use any powerful results about the distribution of prime numbers.) 1

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AwesomeMath 2007 Track 4 — Modulo Arithmetic Week 2 Problem Solving Session July 24, 2007 1. Let a, b be positive integers. Show that gcd( a, b ) lcm( a, b ) = ab . 2. Let a, b, c be positive integers. Show that a divides bc if and only if a gcd( a,b ) divides c . 3. Show that the fraction 21 n + 4 14 n + 3 is irreducible for all positive integers n . 4. Let n be a positive integer. Find gcd( n ! + 1 , ( n + 1)!). 5. Find all positive integers d such that d divides both n 2 + 1 and ( n + 1) 2 + 1 for some integer n . 6. Let a and b be positive integers such that a | b 2 , b 2 | a 3 , a 3 | b 4 , b 4 | a 5 , . . . . Prove that a = b . 7. Let n 2 and k be positive integers. Prove that ( n - 1) 2 | ( n k - 1) if and only if ( n - 1) | k . 8. (AIME 1986) What is the largest positive integer n for which n 3 + 100 is divisible by n + 10? 9. Let m and n be positive integers. Show that (2 m )!(2 n )! ( m + n )! m ! n ! is an integer. 10. Prove that n 2 + 3 n + 5 can never be a multiple of 121 if n is a positive integer. 11. Let a and b > 2 be positive integers. Show that 2 a + 1 is not divisible by 2 b - 1. 12. Let a, b, n > 1 be positive integers. Show that a n + b n is not divisible by a n - b n . 13. Prove that if m > n then a 2 n + 1 is a divisor of a 2 m - 1. Show that if a, m, n are positive with m 6 = n , then gcd( a 2 m + 1 , a 2 n + 1) = ( 1 if a is even, 2 if a is odd. 14. Let n > 1 be a positive integer. Show that 1 + 1 3 + 1 5 + · · · 1 2 n - 1 is not an integer. 2
AwesomeMath 2007 Track 4 — Modulo Arithmetic Week 2 Lecture 7 : Fermat, Euler, and Wilson Yufei Zhao July 25, 2007 Notation: Let Z n = { 0 , 1 , 2 , . . . , n - 1 } denote the complete residue system mod n , and let Z * n = { d | d Z n , gcd( d, n ) = 1 } denote the reduced residue system mod n .

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mod2 - AwesomeMath 2007 Track 4 Modulo Arithmetic Week 2...

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