# 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} = 4n^{2} + 4n + 1 = 2 (2n^{2} + 2n) + 1

Let N = 2n^{2} + 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.

## More references and links

Prime Factors .Divisibility Rules

Divisibility Questions With Solutions