Odd Numbers
- What are Odd Numbers?
- How Do We Know That a Number Is Odd or Even?
- List of Odd Numbers from `1` to `100`
- Types of Odd Numbers
- Properties of Odd Numbers
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What are Odd Numbers?
Odd numbers are those numbers that can't be divided by `2` evenly. Imagine trying to split them into two equal groups – it just doesn't work out neatly. You'll always have something left over. For example, `9` is an odd number because if you divide it by `2`, you get `4` with `1` left over.
The last digit of an odd number is `1`, `3`, `5`, `7`, or `9`. So, `1`, `3`, `5`, `7`, `9`, `11`, `13`, `15`, and so on are all odd numbers. Hence, an odd number can be defined as an integer that is not evenly divisible by `2`.
How Do We Know That a Number Is Odd or Even?
Odd numbers can't be divided evenly into two equal parts. Take number `11`, for example. You can't find two numbers that add up to `11` and are the same. Even numbers are like twins - you can always split them into two equal parts. Think of the number `10`, for instance. You can divide it into `5` and `5`. So, if a number can be evenly split into two equal parts, it's even; otherwise, it's odd.
It is worth noting that an odd number is always one more than or `1` less than an even number.
Below is a list of odd numbers from `1` to `100`:
List of Odd Numbers from `1` to `100`
These are all the numbers in the given range that can't be evenly divided by `2`.
Types of Odd Numbers
Odd numbers fall under various categories based on their characteristics and properties. Some common types include
Composite Odd Numbers:
Composite odd numbers are odd numbers that are not prime. They are formed by multiplying two smaller positive odd integers together. For example, let's consider the odd number `15`. It is composite because it can be expressed as the product of two smaller odd numbers: `3` and `5`. Another example is `21`, which can be written as `3 × 7`.
Consecutive Odd Numbers:
Consecutive odd numbers are pairs of odd numbers that follow each other in sequential order, with a difference of two. For instance, if we start with the odd number `7`, the next consecutive odd number would be `9` because it follows `7` and is two more. Similarly, `11` and `13` are consecutive odd numbers because they are both odd numbers and are two units apart. Another example is `17` and `19`, which also follow each other consecutively with a difference of two.
Properties of Odd Numbers
Properties of Addition:
- When you add an even number with an odd number, you'll always get an odd number. For instance, `9 + 4 = 13`.
- Adding two odd numbers together results in an even number. For example, `7 + 3 = 10`.
Properties of Subtraction:
- Subtracting an odd from an even number gives an odd number. For instance, `12 - 3 = 9`.
- When you subtract two odd numbers, you get an even number. For example, `9 - 5 = 4`.
Properties of Multiplication:
- Multiplying an even number with an odd number or vice versa always gives an even number. For example, `5 × 6 = 30`.
- If you multiply two odd numbers together, the result is always odd, like `7 × 9 = 63`.
Properties of Division:
- Dividing two odd numbers where the denominator is a factor of the numerator always gives an odd number. For example, `15 ÷ 3 = 5`.
- When you divide two odd numbers and the denominator is not a factor of the numerator, the result is a decimal number.
The properties of odd numbers can be summarized using the following table.
Solved Examples
Example `1`. Calculate \( 9 + 7 \). Is the sum an even or an odd number?
Solution:
Since both \( 9 \) and \( 7 \) are odd numbers, according to the property "`"Odd number" + "Odd number" = "Even number"`", their sum is an even number.
\( 9 + 7 = 16 \)
Example `2`. Find \( 12 - 5 \). Is the difference an even or an odd number?
Solution:
\( 12 \) is an even number and \( 5 \) is an odd number. As per the property "`"Even number" - "Odd number" = "Odd number"`", their difference is an odd number.
\( 12 - 5 = 7 \)
Example `3`. Determine \( 11 \times 9 \). Is the product an even or an odd number?
Solution:
Both \( 11 \) and \( 9 \) are odd numbers. As stated by the property "`"Odd number" × "Odd number" = "Odd number"`", their product is an odd number.
\( 11 \times 9 = 99 \)
Example `4`. Compute \( 15 \div 3 \). Is the quotient an even or an odd number?
Solution:
Both \( 15 \) and \( 3 \) are odd numbers, and \( 3 \) is a factor of \( 15 \). According to the property "`"Odd number" ÷ "Odd number" = "Odd number"`", their division results in an odd number.
\( 15 \div 3 = 5 \)
Q`5`. What is the consecutive odd number next to `53`?
Solution:
The consecutive odd number next to `53` is `55`. Both `53` and `55` are odd numbers and they are `2` units apart.
Q`6`. Is `57` a composite odd number?
Solution:
Yes, `57` is a composite odd number. The factors of `57` are `1`, `3`, `19` and `57`. As it has more than `2` factors (`1` and the number itself), `57` is a composite number. Additionally, `57` is an odd number, as `57` ends with `7`.
Practice Problems
Q`1`. Complete the following statement: `"Odd number" + "Odd number" = "_____"`.
- Odd number
- Even number
- Prime number
- Fraction
Answer: b
Q`2`. Is `53` a composite odd number?
- Yes
- No
Answer: b
Q`3`. What is the consecutive odd number after `57`?
- `55`
- `58`
- `56`
- `59`
Answer: d
Q`4`. Fill in the blank: `"Odd number" × "Even number" = "_____"`.
- Odd number
- Even number
- Prime number
- Composite number
Answer: b
Q`5`. Will the result of \( 35 \times 7 \) be an odd or an even number?
- odd number
- even number
Answer: a
Frequently Asked Questions
Q`1`. What is an odd number?
Answer: An odd number is any integer that cannot be divided evenly by `2`. In other words, when divided by `2`, there is always a remainder of `1`.
Q`2`. How do I identify an odd number?
Answer: To identify an odd number, look at the last digit. The number is odd if its last digit is `1`, `3`, `5`, `7`, or `9`. Alternatively, if the number can't be divided evenly by `2`, it's also odd.
Q`3`. What are some properties of odd numbers?
Answer: Odd numbers have several properties, including:
- Adding two odd numbers always results in an even number.
- Subtracting one odd number from another odd number results in an even number.
- Multiplying two odd numbers always results in an odd number.
- Dividing one odd number by another odd number always results in an odd number.
Q`4`. Are there any patterns in the sequence of odd numbers?
Answer: Yes, there are patterns in the sequence of odd numbers. For example, every odd number is preceded and followed by an even number. Additionally, consecutive odd numbers always have a difference of `2` between them.
Q`5`. Can odd numbers be prime numbers?
Answer: Yes, odd numbers can be prime numbers. All prime numbers except for `2` are odd. Prime numbers are integers greater than `1` with no positive divisors other than `1` and themselves, and odd prime numbers follow this rule while being odd.