Factors and Multiples
Factors
A factor of a number is a number that can be multiplied with another number to give the original number.
For example:
– For the number 12, the numbers that multiply together to give 12 are:
1 x 12 = 12, 2 x 6 = 12,3 x 4 = 12
– So, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Steps to Find Factors:
1. Start with 1 and the Number Itself: Every number has 1 and itself as factors.
– Example: The number 18 has 1 and 18 as factors because:
1 x 18 = 18
2. Find Other Pairs of Numbers: Check other smaller numbers to see if they divide the number evenly (without remainder).
– Example for 18:
2 x 9 = 18, 3 x 6 = 18
– So, the factors of 18 are 1, 2, 3, 6, 9, and 18.
Example: Find the Factors of 24
– Start with 1 and 24:
1 x 24 = 24
– Check 2:
2 x 12 = 24
– Check 3:
3 x 8 = 24
– Check 4:
4 x 6 = 24
– So, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Important Points About Factors:
1. Every number has at least two factors: 1 and the number itself.
2. Factors come in pairs: Each factor pair multiplies to give the original number.
3. Smaller numbers may have fewer factors: For example, the factors of 7 are only 1 and 7 because 7 is a prime number (a number with only two factors: 1 and itself).
4. A number can be a factor of multiple numbers: For example, 2 is a factor of 4, 6, 8, 10, and so on.
Examples of Factors:
– Factors of 15: 1, 3, 5, 15
– Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
– Factors of 50: 1, 2, 5, 10, 25, 50
Properties of factors
Properties of factor are-
1. Every Number Has at Least Two Factors:
– Explanation: Every number has 1 and the number itself as factors.
– Example: The number 10 has 1 and 10 as factors because:
1 x 10 = 10
Similarly, for 15, the factors are 1 and 15.
2. Factors Are Always Less Than or Equal to the Number:
– Explanation: A factor of a number is always less than or equal to that number. This is because factors divide the number exactly, so they can’t be larger than the number itself.
– Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. All of these numbers are smaller than or equal to 24.
3. A Number Is Divisible by Its Factors:
-Explanation: A number is divisible by any of its factors without leaving a remainder.
– Example: 12 is divisible by its factors 1, 2, 3, 4, 6, and 12. Dividing 12 by any of these factors gives a whole number:
12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3
No remainder is left.
4. Factors Come in Pairs:
– Explanation: Factors always come in pairs. If you multiply two factors together, you get the original number. These pairs can be different or the same.
– Example: The number 18 has factor pairs:
1 x 18, 2 x 9, 3 x 6
So, the factors of 18 are 1, 2, 3, 6, 9, and 18.
5. 1 Is a Factor of Every Number:
– Explanation: The number 1 divides every number exactly, so 1 is a factor of all numbers.
– Example: For any number like 5, 20, or 100, the number 1 is always a factor:
1 x 5 = 5, 1 x 20 = 20, 1 x 100 = 100
6. The Greatest Factor of a Number Is the Number Itself:
– Explanation: The largest factor of any number is always the number itself.
– Example: The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The largest factor is 36.
7. Prime Numbers Have Only Two Factors:
– Explanation: A prime number has exactly two factors: 1 and the number itself.
– Example: The prime numbers 2, 3, 5, 7, and 11 have only these two factors:
– For 5: Factors are 1 and 5.
– For 7: Factors are 1 and 7.
8. Composite Numbers Have More Than Two Factors:
– Explanation: A composite number has more than two factors.
– Example: The number 12 is composite because it has the factors 1, 2, 3, 4, 6, and 12 (more than two factors).
9. Factors of Zero:
– Explanation: Zero has no factors. Any number divided by zero is undefined, so 0 doesn’t have factors like other numbers.
10. Common Factors:
– Explanation: Sometimes, two or more numbers can have the same factors. These are called common factors.
– Example: The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The common factors of 12 and 18 are 1, 2, 3, and 6.
Common factors
Common factors are numbers that are factors (divisors) of two or more given numbers.
Example:
Let’s find the common factors of 12 and 16.
1. Factors of 12: The numbers that divide evenly into 12 are 1, 2, 3, 4, 6, and 12.
– So, the factors of 12 are: 1, 2, 3, 4, 6, 12
2. Factors of 16: The numbers that divide evenly into 16 are 1, 2, 4, 8, and 16.
– So, the factors of 16 are: 1, 2, 4, 8, 16
Common Factors:
Now, let’s see which numbers are common in both lists:
– Factors of 12: 1, 2, 3, 4, 6, 12
– Factors of 16: 1, 2, 4, 8, 16
The common factors of 12 and 16 are the numbers that appear in both lists: 1, 2, 4.
Example2:
Find the common factors of 18 and 24.
1. Factors of 18: 1, 2, 3, 6, 9, 18
2. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors of 18 and 24: 1, 2, 3, 6
Multiples
Multiples are the result of multiplying a number by whole numbers (1, 2, 3, 4, and so on). In other words, multiples of a number are the numbers you get when you multiply it by different integers.
For example:
– The multiples of 2 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and so on.
– The multiples of 5 are: 5, 10, 15, 20, 25, 30, and so on.
The multiples of 3.
– Multiples of 3: To find multiples of 3, we multiply 3 by different whole numbers.
– 3 x 1 = 3
– 3 x 2 = 6
– 3 x 3 = 9
– 3 x 4 = 12
– 3 x 5 = 15
– and so on…
So, the first few multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and they continue infinitely.
Properties of Multiples
Properties of Multiples
1. Every Number is a Multiple of Itself
– The smallest multiple of any number is the number itself. This is because any number multiplied by 1 gives the number.
Example:
– Multiples of 5: 5, 10, 15, 20, 25
– Here, the smallest multiple of 5 is 5 itself.
2. Multiples are Infinite- Multiples of a number go on forever. There’s no limit to how many multiples a number can have because you can keep multiplying by larger numbers.
Example:
– Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24…
– The multiples will continue without end.
3. Every Multiple is Greater Than or Equal to the Number
– Multiples of a number are always equal to or greater than the number. The first multiple of any number is the number itself, and then it gets bigger as you multiply by larger numbers.
Example:
– Multiples of 4: 4, 8, 12, 16, 20
– Each multiple is greater than or equal to 4.
4. Zero is a Multiple of Every Number
– If you multiply any number by 0, the result is always 0. So, 0 is considered a multiple of every number.
Example:
– 5 x 0 = 0
– 8 x 0 = 0
– So, 0 is a multiple of both 5 and 8.
5. Multiples of a Number are Divisible by That Number
– If you take any multiple of a number, that multiple can be divided evenly by the original number.
Example:
– Multiples of 6: 6, 12, 18, 24
– All of these numbers can be divided evenly by 6:
6 ÷ 6 = 1 , 12 ÷ 6 = 2 \), 18 ÷ 6 = 3 , 24 ÷ 6 = 4
6. Multiples of 2 are Called Even Numbers
– All multiples of 2 are called even numbers because they can be divided evenly by 2.
Example:
– Multiples of 2: 2, 4, 6, 8, 10
– These are all even numbers.
Common Multiples
Common multiples are multiples that two or more numbers share. In other words, if you list the multiples of two or more numbers, the common multiples are the numbers that appear in all lists.
How to Find Common Multiples:
1. First, find the multiples of the two numbers.
2. Then, identify the multiples that are the same in both lists.
Example 1:
Let’s find the common multiples of 3 and 4.
1. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, …
2. Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, …
Now, let’s find the common multiples. The numbers that appear in both lists are:
– 12, 24, 36, and so on.
So, the common multiples of 3 and 4 are 12, 24, 36, and so on.
Example 2:
Let’s find the common multiples of 5 and 6.
1. Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, …
2. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, …
Common multiples of 5 and 6:
– 30, 60, and so on.
Prime and Composite numbers
What Are Prime Numbers?
A prime number is a number that has exactly two factors: 1 and the number itself.
– In other words, prime numbers can only be divided evenly by 1 and by the number itself.
Examples of Prime Numbers:
– 2 is a prime number because it has only two factors: 1 and 2.
– 3 is a prime number because it has only two factors: 1 and 3.
– 5 is a prime number because it has only two factors: 1 and 5.
First Few Prime Numbers:
The first few prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31…
What Are Composite Numbers?
A composite number is a number that has more than two factors. This means that composite numbers can be divided by 1, the number itself, and at least one other number.
Examples of Composite Numbers:
– 4 is a composite number because it has three factors: 1, 2, and 4.
– 6 is a composite number because it has four factors: 1, 2, 3, and 6.
– 9 is a composite number because it has three factors: 1, 3, and 9.
First Few Composite Numbers:
The first few composite numbers are:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20…
Special Case: The Number 1
– 1 is neither a prime nor a composite number because it has only one factor: 1.
Prime factorization
Prime factorization is the process of breaking down a number into its prime factors—the prime numbers that multiply together to give the original number.
Steps for Prime Factorization:
1. Start with the smallest prime number (which is 2) and check if it divides the given number evenly.
2. Keep dividing the number by prime numbers (like 2, 3, 5, 7, etc.) until you cannot divide anymore.
3. The numbers you multiply together at the end are the prime factors of the original number.
Example 1: Prime Factorization of 12
Let’s find the prime factorization of 12.
1. Start with 2, the smallest prime number.
– 12 ÷ 2 = 6 (12 can be divided by 2)
2. Now, divide 6 by 2 again.
– 6 ÷ 2 = 3
3. 3 is a prime number, so we stop here.
So, the prime factorization of 12 is:
12 = 2 x 2 x 3
Or you can write it as:
12 = 2^2 x 3
This means that 12 is made up of two 2’s and one 3.
Example 2: Prime Factorization of 18
Let’s find the prime factorization of 18.
1. Start with 2.
– 18 ÷ 2 = 9
2. Now, try dividing 9 by 2. It doesn’t work, so move to the next prime number, 3.
– 9 ÷ 3 = 3
3. Divide the result 3 by 3 again.
– 3 ÷ 3 = 1
So, the prime factorization of 18 is:
18 = 2 x 3 x 3
Or you can write it as:
18 = 2 x 3^2
This means that 18 is made up of one 2 and two 3’s.
Factor tree method and division method
There are two common ways to find the prime factorization of a number:
1. Factor Tree Method
2. Division Method
1. Factor Tree Method
In the Factor Tree Method, we break a number down step-by-step into factors until all the factors are prime numbers. We represent this process using a “tree” structure.
Steps:
1. Start by writing the number at the top.
2. Find any two numbers that multiply together to give that number.
3. Keep breaking down each number into factors until you can’t factor anymore. The remaining numbers should be prime.
Example: Prime Factorization of 24 Using Factor Tree
Let’s find the prime factorization of 24.
1. Start with 24. We can split 24 into 4 × 6.
– 24
↙ ↘
4 6
2. Now, split 4 into 2 × 2, and 6 into 2 × 3. (These are all prime numbers.)
– 24
↙ ↘
4 6
↙ ↘ ↙ ↘
2 2 2 3
3. We can stop here because all the numbers at the bottom are prime (2 and 3).
So, the prime factorization of 24 is:
24 = 2 x 2 x 2 x 3
Or:
24 = 2^3 x 3
2. Division Method
In the Division Method, we repeatedly divide the number by prime numbers, starting with the smallest prime (which is 2), until the result is 1.
Steps:
1. Start by dividing the number by the smallest prime number (like 2).
2. Keep dividing the result by prime numbers until you reach 1.
3. The divisors you used are the **prime factors**.
Example: Prime Factorization of 24 Using Division Method
1. Start with 24 and divide by 2 (the smallest prime number):
– 24 ÷2 = 12
2. Divide 12 by 2 again:
– 12 ÷ 2 = 6
3. Divide 6 by 2:
– 6 ÷ 2 = 3
4. Now divide 3 by 3 (since 3 is the next prime number):
– 3 ÷ 3 = 1
Now that we’ve reached 1, the divisors (prime numbers) are:
24 = 2 x 2 x 2 x 3
Or:
24 = 2^3 x 3
Tests of divisibility
Divisibility rules help us quickly determine whether one number can be divided by another without actually doing the division. These rules are especially useful when you’re working with large numbers. Let’s explore the divisibility rules for common numbers like 2, 3, 4, 5, 6, 9, and 10.
1. Divisibility by 2
A number is divisible by 2 if it is an even number, meaning its last digit is 0, 2, 4, 6, or 8.
Example:
– 48 is divisible by 2 because the last digit is 8 (an even number).
– 57 is not divisible by 2 because the last digit is 7 (an odd number).
2. Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example:
– For 123, sum of digits: 1 + 2 + 3 = 6 . Since 6 is divisible by 3, 123 is divisible by 3.
– For 142, sum of digits: 1 + 4 + 2 = 7 . Since 7 is not divisible by 3, 142 is not divisible by 3.
3. Divisibility by 4
A number is divisible by 4 if the last two digits form a number that is divisible by 4.
Example:
– For 516, check the last two digits (16). Since 16 is divisible by 4, 516 is divisible by 4.
– For 732, check the last two digits (32). Since 32 is divisible by 4, 732 is divisible by 4.
4. Divisibility by 5
A number is divisible by 5 if its last digit is either 0 or 5.
Example:
– 75 is divisible by 5 because the last digit is 5.
– 82 is not divisible by 5 because the last digit is 2.
5. Divisibility by 6
A number is divisible by 6 if it is divisible by both 2 and 3.
Example:
– 108: The last digit is 8 (so it’s divisible by 2), and the sum of the digits is 1 + 0 + 8 = 9 (which is divisible by 3). Therefore, 108 is divisible by 6.
– 124: The last digit is 4 (so it’s divisible by 2), but the sum of digits is 1 + 2 + 4 = 7 (which is not divisible by 3). So, 124 is not divisible by 6.
6. Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example:
– For 189, sum of digits: 1 + 8 + 9 = 18 . Since 18 is divisible by 9, 189 is divisible by 9.
– For 123, sum of digits: \( 1 + 2 + 3 = 6 \). Since 6 is not divisible by 9, 123 is not divisible by 9.
7. Divisibility by 10
A number is divisible by 10 if its last digit is 0.
Example:
– 230 is divisible by 10 because the last digit is 0.
– 145 is not divisible by 10 because the last digit is 5.
Quick Recap of Divisibility Rules:
1. Divisible by 2: Last digit is 0, 2, 4, 6, or 8.
2. Divisible by 3: Sum of digits is divisible by 3.
3. Divisible by 4: Last two digits form a number divisible by 4.
4. Divisible by 5: Last digit is 0 or 5.
5. Divisible by 6: Divisible by both 2 and 3.
6. Divisible by 9: Sum of digits is divisible by 9.
7. Divisible by 10: Last digit is 0.