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

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.

- Natural Number – It contains numbers starting from 1.
- Whole Number – It contains numbers starting from 0.
- Even Number – Numbers that are divisible by 2.
- Odd Number – Numbers that are not divisible by 2.
- Prime Number – Numbers that are divisible by 1 and itself only. i.e. only two factors
- 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,

number | 1st group | 2nd group | Equality |
---|---|---|---|

2 | 1 | 1 | yes |

4 | 2 | 2 | yes |

6 | 3 | 3 | yes |

8 | 4 | 4 | yes |

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.

num1 | num2 | num1+num2 |
---|---|---|

2 | 8 | 10 |

12 | 16 | 28 |

- The sum of two odd numbers is always even.

num1 | num2 | num1+num2 |
---|---|---|

5 | 5 | 10 |

7 | 9 | 16 |

- When even and odd numbers are multiplied, the result is always even.

num1 | num2 | num1 × num2 |
---|---|---|

9 | 8 | 72 |

3 | 6 | 18 |

- When even numbers are divided by 2, the remainder is always zero.

num1 | num2 | remainder of num1 ÷ num2 |
---|---|---|

12 | 2 | 0 |

80 | 2 | 0 |

### 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

bwhich 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 =

eveni.e. 2b =

odda + b = even + odd = odd

(from property)Hence, it is odd.

Second argument:

a =

oddb =

odda + 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 SystemThis number system contains the digits or numbers having base 2 i.e. only 0 and 1. For example, 1001

_{2 }is a binary number.

2. Octal Number SystemThis number system contains the digits or numbers starting from 0 to 7 and has a base 8. For example, 242

_{8 }is an octal number.

3.Decimal Number SystemThis number system contains the digits or numbers starting from 0 to 9 and has a base of 10. For example, 1024

_{10}is a decimal number.

4. Hexadecimal Number SystemThis 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, A45B

_{16}is an hexadecimal number.

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