Table of contents | |
Even Numbers | |
Odd Numbers | |
Consecutive Integers | |
Even and Odd Number Properties | |
Arithmetic Rules of Odds & Evens | |
Other Important Properties of Numbers |
Even and odd numbers are two categories of integers, which are whole numbers and their negatives, including zero. Understanding the characteristics of even and odd numbers is fundamental in mathematics.
Now that we know what are even and odd numbers, let’s see their properties.
Odd ± Even = Odd
Odd x Odd = Odd
Odd + Odd = Even
Even x Even = Even (and divisible by 4)
Even ± Even = Even
Odd x Even = Even
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?
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.
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
Step 2: Round it off to the closest integer
Step 3: List down all the prime numbers which are less or equal to this integer
Step 4: Check whether any of these prime numbers can divide the given number or not
Step 5: If yes then the given number is not a prime number, else it is a prime number
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.
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
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.
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
To find
Approach and Working out
Now, if we see, in the given expression
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.
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
To find
Approach and Working out
The value of p = 2, since 2 is the only even prime number
And, we know that,
Therefore, the value of q = 42/2 = 21.
Hence, the correct answer is Option D.
110 videos|110 docs|120 tests
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1. What are even numbers? |
2. What are odd numbers? |
3. What are consecutive integers? |
4. What are the properties of even numbers? |
5. What are the properties of odd numbers? |
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