the division algorithm The fact that we can divide integers and get a unique quo-tient and remainder is the key to understanding divisibility, congruence, and modular arithmetic Theorem 17 13 (The Division Algorithm) Let n;d2Zwith d6= 0 Then there are unique integers q;rsuch that n= qd+r and 0 r
CCBC Math 081 Introduction to Integers Section 1 1 Third Edition 10 pages 2 1 1 Introduction to Integers There once was a line full of numbers That never did sleep nor did slumber There are many good reasons To compute in all seasons In Fall, Winter, Spring, and the Summer This little known limerick by an even lesser known
18 There are 30students in Mr Moore’s discrete mathematics class: 23of his students major in mathematics, 12major in physics, and 3major in subjects outside of math and physics How many double major in mathematics and physics? A 2 B 3 C 5 D 7 E 8 19 How many 0’s appear at the end of the product of the ?rst 2018 primes? A 0 B 1 C 22 D 3
2) How many integers are there between ±8 and 2? a) 7 b) 4 c) 0 d) 9 3) What is the opposite value of the integer 6? a) ±6 b) 5 c) 6 d) ±4 5) Which of the following integers is greater than ±1 and lesser than 7? a) ±9 b) 5 c) ±5 d) 8 6) How many pairs of opposite integers are there between ±4 and 5? a) 3 b) 8 c) 2 d) 6
There is a positive constant csuch that if nis su ciently large, m>c(logn)2, and I;Jˆ[n] are arithmetic progressions of length mwith common di erence 1 or 2, the following holds Whenever SˆIand TˆJare nonempty, with jSj+ jTj m, there is a 2-coprime pair s;twith s2S, t2T Corollary 1 There is a bijection of I and J with corresponding
Corollary 2 2 The positive integers a and b are relatively prime i? there exist integers x and y such that ax+by = 1 Corollary 2 3 For any positive integers a 1, a 2, , a n, there exist integers x 1, x 2, , x n, such thata 1x 1+a 2x 2+···+a nx n = gcd(a 1,a 2, ,a n) Corollary 2 4 Let a and b be positive integers, and let n be
Problem 1: (Section 6 1 Exercise 14) How many integers between 1 and 1000 the number of elements in the union is the sum of the elements in each set
Rational Numbers These are any numbers that can be expressed as a fraction, which includes all integers and most decimals Examples include - 1 2 , 208,
How many positive integers less than 1000 As shown in (a), 142 integers are divisible by 7 So there are 90 − 9 = 81 2-digit integers with distinct digits