# Properties of Numbers – Even & Odd | Prime | HCF & LCM

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:

• 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 not a multiple of 2 is an odd number
• 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.

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 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

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## 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 112 = 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 = 22 × 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 = 23 × 32
• 300 = 22 × 3 × 52

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 = 23 × 32 × 52

### 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 = 23 × 32
• 300 = 22 × 3 × 52

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 = 22 × 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, 2k2 + 14k + 7, where k is a positive integer?

1. Even
2. Odd
3. Cannot be determined

### Solution

Given

• An e×pression, 2k2 + 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, 2k2 + 14k + 7 = 2k2 + 14k + 6 + 1

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

• The term 2k2 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, 2k2 + 14k + 7 = 2k2 + 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?

1. 6
2. 7
3. 14
4. 21
5. 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. Payal Tandon 