# 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.