Hello Reader,

We see that you want to know the basics and properties of numbers.

*You have landed just at the right place. *

## Purpose of the article:

In this article,

- You will get a deeper idea about the behavior of numbers and its operations
- You will also get to know a few facts about the properties of the numbers.

So, let’s get straight into the article.

# Properties of numbers: Even – Odd

## Even Numbers

- Any integer that is a multiple of 2 is an even number.
- So, any integer which is in the form of 2k, where k is an integer, is an even number
- So, all integers that end with 0, 2, 4, 6 or 8 are even numbers

__E×ample__: 4, 56, 98, 200 are all even numbers

## Odd Numbers

- Any integer which is not an even number is an odd number
- Or, in other words, an integer that is
a multiple of 2 is an odd number__not__ - So, any integer which is in the form of 2k ± 1, where k is an integer, is an odd number
- So, all integers that end with 1, 3, 5, 7 or 9 are odd numbers

__Example__: 7, 31, 75, 499 are all odd numbers

**Note: **There is an odd number between every two consecutive even numbers.

According to the definitions given above is 0 an even or an odd number?

## Properties of numbers: Even and Odd number properties

Now that we know what are even and odd numbers, let’s see their properties.

Number 1 | Operation | Number 2 | | Outcome | Examples |

Even | ± | Even | = | Even | · 4 + 2 = 4 · 4 – 2 = 2 |

Even | ± | Odd | = | Odd | · 4 + 1 = 5 · 4 – 1 = 3 |

Odd | ± | Even | = | Odd | · 5 + 2 = 7 · 5 – 2 = 3 |

Odd | ± | Odd | = | Even | · 5 + 3 = 8 · 5 – 3 = 2 |

Even | × | Even | = | Even | · 2 × 4 = 8 |

Even | × | Odd | = | Even | · 2 × 3 = 6 |

Odd | × | Even | = | Even | · 3 × 2 = 6 |

Odd | × | Odd | = | Odd | · 3 × 5 = 15 |

Even | ÷ | Even | = | May or may not be an integer | · 4 ÷ 2 = 2 · 2 ÷ 4 = 0.5 |

Even | ÷ | Odd | = | May or may not be an integer | · 4 ÷ 1 = 4 · 4 ÷ 3 = 1.33 |

Odd | ÷ | Even | = | Not an integer | · 3 ÷ 2 = 1.5 |

Odd | ÷ | Odd | = | May or may not be an integer | · 3 ÷ 1 = 3 · 1 ÷ 3 = 0.33 |

## Properties of Numbers: Prime Numbers

### Definition

- A number which is divisible by only two number, 1 and the number itself, is called a prime number

__Example__: 2, 3, 5, 7, 11……….

**Note:** 2 is the only prime number, which is even, because all other even numbers will be divisible by at least three numbers, 1, 2 and the number itself.

- Every positive integer can be expressed as a product of one or more prime numbers

__Example__: 55 = 5 * 11, where 5 and 11 are two prime numbers

Based on the above definition, Is 1 a prime number?

*So, we know what is a prime number, but how do we check whether a given number is prime or not?*

## What is the process to check whether a given number is prime or not?

It is easy to check whether a single-digit or a two-digit number is prime or not, but what if we are given a three-digit number or more.

*Let’s say the number is 123, can we quickly find whether 123 is prime or not?*

For this, let us learn a five-step approach to check whether a given number is a prime number or not?

**Step 1**: *Find the square root of the given number*

- The square root of 123 approximately equal to 11 as 11
^{2}= 121

**Step 2**: *Round it off to the closest integer*

- The closest integer is 11.

**Step 3**: *List down all the prime numbers which are less or equal to this integer*

- Prime numbers less than or equal to 11 are 2, 3, 5, 7, and 11.

**Step 4**:* Check whether any of these prime numbers can divide the given number or not *

- 123 is not divisible by 2.
- But 123 is divisible by 3.
- We need not check for other prime numbers

**Step 5**: *If yes then the given number is not a prime number, else it is a prime number*

- 123 is divisible by 3. Hence, it is not a prime number.

Pop-Quiz: is 157 a prime number?

## Prime factorization

We already know that any given positive integer can be expressed as a product of one or more prime numbers. This representation of any number is called prime factorization.

__For example__:

420 = 2 × 210 = 2 × 2 × 105 = 2 × 2 × 3 × 35 = 2 × 2 × 3 × 5 × 7 = 2^{2} × 3 × 5 × 7

Now, let us learn a few more properties of numbers where we apply prime factorization.

## Properties of numbers: Least Common Multiple (LCM)

Before we see what an LCM is, let us understand the meaning of a multiple.

** Multiple:** If the remainder when a number “N” is divided by another number “n” is zero,

*then N is said to be a multiple of n*.

And, LCM is the smallest common multiple of any two or more given positive integers.

__For example__: LCM of 4 and 6 is 12, since 12 is the smallest number, which is a multiple of both 4 and 6

## Properties of numbers: Highest Common Factor or Greatest Common Divisor (HCF/GCD)

Let us first learn what is a factor or divisor, and then we can learn about GCD or HCF.

** Factor/Divisor: **If the remainder when a number “N” is divided by another number “n” is zero, then

*n is said to be a factor or divisor of N*.

And, HCF or GCD is the largest common factor/divisor to any two or more given positive integers.

* For example:* The GCD of 24 and 30 is 6, since 6 is the greatest number, which is a factor of both 24 and 30.

## Method to find LCM and GCD of any two given positive integers

Now that we know what is an LCM and GCD, let us learn a method to find the LCM and GCD of any two given positive integers.

### How to find LCM?

Let us consider two positive integers, 72 and 300

**Step 1**: *Represent the two given numbers in their prime factorization form*

- 72 = 2
^{3}× 3^{2} - 300 = 2
^{2}× 3 × 5^{2}

**Step 2**: *List down all the distinct prime factors from both the numbers*

- Prime factors of 72 = 2 and 3
- Prime factors of 300 = 2, 3 and 5
- Thus, distinct prime factors from both combined are 2, 3 and 5

**Step 3**: *Find the power of each of these prime factors, which is the greatest between both the numbers.*

- Power of 2 in 72 = 3, and power of 2 in 300 = 2,
- So, the power of 2 in LCM of 72 and 300 = 3, since the power of 2 is greatest in 72
- Similarly, we get the powers of 3 and 5 as 2 and 2 respectively

**Step 4**: *Multiply all these terms to get the LCM*

- Therefore, the LCM of 72 and 300 = 2
^{3}× 3^{2}× 5^{2}

### How to find GCD/HCF?

Let us consider the same two positive integers, 72 and 300

**Step 1**: *Represent the two given numbers in their prime factorization form*

- 72 = 2
^{3}× 3^{2} - 300 = 2
^{2}× 3 × 5^{2}

**Step 2**: *List down all the common prime factors*

- Prime factors of 72 = 2 and 3
- Prime factors of 300 = 2, 3 and 5
- Thus, common prime factors 2 and 3

**Step 3**: *Find the power of each of these prime factors, which is the smallest between both*

- Power of 2 in 72 = 3, and power of 2 in 300 = 2,
- So, the power of 2 in GCD of 72 and 300 = 2, since the power of 2 is the smallest in 300
- Similarly, we get the powers of 3 and 5 as 1 and 0 respectively

**Step 4**: *Multiply all these terms to get the GCD*

- Therefore, the GCD of 72 and 300 = 2
^{2}× 3 = 12

__Note:__*The product of LCM and GCD of any two given positive integers is equal to the product of the two given integers.*

*LCM × HCF = Product of the integers*

What is the LCM and GCD of two prime numbers? If you liked this article here are a couple of more article related to number properties:

## Properties of Numbers: Application Quiz

### Question 1

What is the even/odd nature of the expression, 2k^{2} + 14k + 7, where k is a positive integer?

- Even
- Odd
- Cannot be determined

### Solution

**Given**

- An e×pression, 2k
^{2}+ 14k + 7, where k is a positive integer

**To find**

- The even/odd nature of the given e×pression

**Approach and Working out**

- We can write, 2k
^{2}+ 14k + 7 = 2k^{2}+ 14k + 6 + 1

Now, if we see, in the given e×pression

- The term 2k
^{2}is always even, for any value of k - Since, even * any integer = even
- The term 14k is always even, for any value of k
- Since, even * any integer = even
- The term 6 is also even, and 1 is odd

So, 2k^{2} + 14k + 7 = 2k^{2} + 14k + 6 + 1 = even + even + even + odd = odd

Therefore, the given expression is always odd, for any value of k

Hence, the correct answer is **Option B**.

## Properties of Numbers Quiz: Question 2

The LCM of two positive integers, p, and q, is 42 and HCF is 1. If p is an even prime number, then what is the value of q?

- 6
- 7
- 14
- 21
- 42

### Solution

**Given**

- Two positive integers, p, and q
- p is an even prime number
- LCM pf p and q = 42
- HCF of p and q = 1

**To find**

- The value of q

**Approach and Working out**

The value of p = 2, since 2 is the only even prime number

And, we know that,

- LCM × HCF = the product of the two integers
- 42 × 1 = 2 × q

Therefore, the value of q = 42/2 = 21.

Hence, the correct answer is **Option D**.

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