  Even and Odd Numbers Problems

Problems and detailed solutions on even and odd numbers are presented. More questions and their solutions are also included.

Definitions

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

2 n

An odd number is any integer not divisible by 2.
Example: ...-5, -3, -1, 1, 3, ...
Any odd number may be written as

2 n + 1

Problems on Even and Odd Numbers

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 2 n and 2 k be the two even numbers. The sum of the two numbers is written in factored form as follows
2 n + 2 k = 2(n + k)
Let N = n + k and write the sum as
2 n + 2 k = 2 N
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 2 n be the even number and 2 k + 1 be the odd number. The sum of the two numbers is given by
(2 n) + (2 k + 1) = 2 n + 2 k + 1 = 2(n + k) + 1
Let N = n + k and write the sum as
(2 n) + (2 k + 1) = 2 N + 1
The sum is an odd number.

Problem 5:
Show that the sum of two odd numbers is an even number.
Solution
Let 2 n + 1 and 2 k + 1 be the odd numbers to add. The sum of the two numbers is given by
(2 n + 1) + (2 k + 1) = 2 n + 2 k + 2 = 2(n + k + 1)
Let N = n + k + 1 and write the sum as
(2 n + 1) + (2 k + 1) = 2 N
The sum is an even number.

Problem 6:
Show that the sum of three odd numbers is an odd number.
Solution
Let 2 m + 1, 2 n + 1 and 2 k + 1 be the odd numbers to add. The sum of the three numbers is given by
(2 m + 1) + (2 n + 1) + (2 k + 1) = 2 m + 2 n + 2 k + 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
(2 n + 1) + (2 m + 1) + (2 k + 1) = 2 N + 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.

Questions

Complete the sentences below 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 of two even numbers is . . .
4. The sum of two even numbers and one odd number is . . .
5. The square of an even number is . . .

Solutions

1. ODD
Solution
Let 2 n + 1 and 2 p + 1 be two odd numbers
(2 n + 1 )(2 p + 1) = (2 n)(2 p) + 2 p + 2 n + 1 = 2 (2 n p + p + n) + 1 , which is an odd number.

2. EVEN
Solution
Let 2 n and 2 p be two even numbers
(2 n )(2 p) = (2 n)(2 p) = 2 (2 n p ) , which is an even number.

3. EVEN
Solution
Let 2 n and 2 p be two even numbers
2 n - 2 p = 2 (n - p) , which is an even number.

4. ODD
Solution
Let 2 n and 2 p be two even numbers and 2 m + 1 be an odd number.
2n + 2 p + 2 m + 1 = 2 (n + p + m) + 1 , which is an odd number.

5. EVEN
Solution
Let 2 n be an even numbers
(2 n)
2 = 2 2 n 2 = 4 n 2 = 2 (2 n 2 ) , which is an even number.