# If two numbers a and b are even, then prove that their sum a + b is even

• Last Updated : 03 Sep, 2021

Number System is a system of representing numbers using symbols and digits in a certain manner. You can think of it as a Grammar book in mathematics. As soon as we hear this word “Number” 1,2,3,… get pop in our head immediately. Number System defines their value, operations to perform, and other properties.

### Types of Numbers

Each number is unique. It has many variations. Thus, it can be considered as a natural number, whole number, even number, odd number, prime number, composite number, etc.

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1. Natural Number – It contains numbers starting from 1.
2. Whole Number – It contains numbers starting from 0.
3. Even Number – Numbers that are divisible by 2.
4. Odd Number – Numbers that are not divisible by 2.
5. Prime Number – Numbers that are divisible by 1 and itself only. i.e. only two factors
6. Composite Number – Numbers that are divisible by 1and itself and others. i.e. more than two factors

### Even numbers

Those numbers which are multiples of 2 are known as Even numbers. e.g. 2, 4, 6, 8, 10, … etc. These numbers can be split into two equal groups or pairs. Odd numbers cannot be split into equal numbers. Let’s try to understand it,

Hence, Even numbers can be split equally. Let’s briefly look at some of the important properties of Even numbers.

Some Properties of Even number

• The sum of two even numbers is always even.
• The sum of two odd numbers is always even.
• When even and odd numbers are multiplied, the result is always even.
• When even numbers are divided by 2, the remainder is always zero.

### If two numbers a and b are even, then prove that their sum a + b is even

Solution:

Suppose,

a = 2 × X

b = 2 × Y

where X and Y are any integers that may be even or odd. Multiplying any number by 2 is always even so 2X and 2Y are even.

a + b = 2 × X + 2 × Y = 2 × (X + Y)

The result has a factor of 2. So, it is always even.

Example:

Input: a = 2, b = 4

Output: a + b = 2 + 4 = 6 = 2 × 3

Hence, it is even.

### Similar Problems

Question 1: If two numbers a and b are even and odd, then their sum a + b is odd.

Solution:

Suppose,

a = 2 × X

b = 2 × Y + 1

where X and Y are any two integers.

a + b = 2 × X + 2 × Y + 1 = 2 × (X + Y) + 1

The result is similar to b which is an odd number. Hence, it is odd.

Example:

Input: a = 4

Input: b = 7

Output: a + b = 4 + 7 = 11

Hence, it is odd.

Question 2: If two numbers are a and b are odd, then their sum a + b is even.

Solution:

Suppose,

a = 2 × X + 1

b = 2 × Y + 1

where X and Y are any two integers

a + b = 2 × X + 1 + 2 × Y + 1 = 2 × X + 2 × Y + 2 = 2 × (X + Y + 1)

Which is a factor of 2, so it is even.

Example:

Input: a = 3

Input: b = 5

Output: a + b = 3 + 5 = 8

Hence it is even.

Question 3: What is the sum of two prime numbers a and b?

Solution:

All the prime numbers are odd except 2 which is even.

Hence, it is concluded that the addition of any two prime numbers is always even if we ignore 2.

First argument:

a = even i.e. 2

b = odd

a + b = even + odd = odd (from property)

Hence, it is odd.

Second argument:

a = odd

b = odd

a + b = odd + odd = even (from property)

Hence, it is even.

Question 4: What are the different types of Number Systems?

Solution:

There are four types of number Systems,

1. Binary Number System

This number system contains the digits or numbers having base 2 i.e. only 0 and 1. For example, 10012 is a binary number.

2. Octal Number System

This number system contains the digits or numbers starting from 0 to 7 and has a base 8. For example, 2428 is an octal number.

3. Decimal Number System

This number system contains the digits or numbers starting from 0 to 9 and has a base of 10. For example, 102410 is a decimal number.

This number system contains the digits or numbers starting from 0 to 15 and has a base of 16.

A – 10

B – 11

C – 12

D – 13

E – 14

F – 15

For example, A45B16 is an hexadecimal number.

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