# Even and Odd Numbers Problems

 Tutorials with problems and detailed solutions on even and odd numbers are presented. Exercies are also included after the tutorial. An even number is any integer divisible by 2. Example: ...-4, -2, 0, 2, 4, ... Any even number may be written as a multiple of 2 as 2n An odd number is any integer not divisible by 2. Example: ...-5, -3, -1, 1, 3, ... Any odd number may be written as 2n+1 Problem 1: List all even numbers greater than -4 and smaller than 20. Solution -2, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18 Problem 2: List all odd numbers greater than 3 and smaller than 30. Solution 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29. Problem 3: Show that the sum of two even numbers is even. Solution Let 2n and 2k be the two even numbers. The sum of the two numbers is written in factored form as follows 2n + 2k = 2(n + k) Let N = n + k and write the sum as 2n + 2k = 2N The sum is an even number. Problem 4: Show that the sum of an even number and an odd number is an odd number. Solution Let 2n be the even number and 2k + 1 be the odd number. The sum of the two numbers is given by (2n) + (2k + 1) = 2n + 2k + 1 = 2(n + k) + 1 Let N = n + k and write the sum as (2n) + (2k + 1) = 2N + 1 The sum is an odd number. Problem 5: Show that the sum of two odd numbers is an even number. Solution Let 2n + 1 and 2k + 1 be the odd numbers to add. The sum of the two numbers is given by (2n + 1) + (2k + 1) = 2n + 2k + 2 = 2(n + k + 1) Let N = n + k + 1 and write the sum as (2n + 1) + (2k + 1) = 2N The sum is an even number. Problem 6: Show that the sum of three odd numbers is an odd number. Solution Let 2m + 1, 2n + 1 and 2k + 1 be the odd numbers to add. The sum of the three numbers is given by (2m + 1) + (2n + 1) + (2k + 1) = 2m + 2n + 2k + 3 = 2(m + n + k) + 2 + 1 = 2(m + n + k + 1) + 1 Let N = n + m + k + 1 and write the sum of the three odd numbers as (2n + 1) + (2m + 1) + (2k + 1) = 2N + 1 The sum of three odd numbers is an odd number. Problem 7: Show that the square of an odd number is an odd number. Solution Let 2n + 1 be the odd number to square and expand the square. (2n + 1)2 = 4n2 + 4n + 1 = 2 (2n2 + 2n) + 1 Let N = 2n2 + 2n and write the square of the odd number as (2n + 1)2 = 2 N + 1 The square of an odd number is an odd number. Problem 8: Show that the product of an odd number and an even number is an even number. Solution Let 2 m + 1 be the odd number and 2n be the even number. The product is given by (2 m + 1)(2 n) = 4mn + 2n = 2(2m n + n) Let N = 2m n + n and write the product as (2 m + 1)(2 n) = 2 N The product of an odd number and an even number is an even number. Exercises: Complete using the word even or the word odd. 1. The product of two odd numbers is . . . 2. The product of two even numbers is . . . 3. The difference between two even numbers is . . . 4. The sum of two even numbers and one odd number is . . . 5. The square of an even number is . . . Answers to Above Exercises. 1. ODD 2. EVEN 3. EVEN 4. ODD 5. EVEN More on numbers and Fractions.