Factors and Multiples

Divisibility rules

A number is completely divisible by another number if remainder is 0.

A whole number Is divisible by

2: if the last digit is 0 or divisible by 2 (even). For example, 54, 98, 22 and 20.

3: if the sum of its digits is divisible by 3. For example, (291) [2 + 9 + 1 = 12]

5: if the last digit is 0 or 5. For example 590, 6815

Is 81 divisible by 3?

Yes, because 8 + 1 = 9 and 9 is divisible by 3.

Recall 3 × 3 = 9.

Is 250 divisible by 5?

Yes, because the last digit is 0

Circle the numbers divisible by 2. 24 32 51 761 882 1000

Circle the numbers divisible by 3. 24 33 81 100 1101 1211

Is 341 divisible by 2? ______________________________________________

 Is 2555 divisible by 5? ______________________________________________

A whole number Is divisible by

4: if the number formed from the last two digits is divisible by 4. For example, 112, 1308, 2520

6: if the number is divisible by both 2 and 3. For example, 12, 24, 114, 3312

10: if the last digit is 0. For example, 110. 28900

Is 1124 divisible by 4?

Yes, because the number formed from the last two digits is divisible by 4.

Recall 4 × 6 = 24

Is 1550 divisible by 10?

Yes, because the last digit is 0

Circle the numbers divisible by 4. 12 35 344 748 4112

 Circle the numbers divisible by 6. 12 33 312 902 3114

Is 341 divisible by 4? ______________________________________________

 Is 2555 divisible by 6? ______________________________________________

Is 2555 divisible by 10? ______________________________________________

Factors and Multiples

Prime and composite numbers

 Prime number is a number that can only be divided exactly by 1 and itself.

Think about number 3.

It can only be divided by 1 and 3, without leaving a remainder.

 3 is a prime number.

Some other prime numbers are 2, 5, 7, 1l and 13 etc.

Composite number is a number that can also be divided exactly by any number other than 1 and itself.

Think about number 4.

 It can be divided by 1, 2.and 4 without leaving a remainder.

4 is a composite number.

Some other composite numbers are 6, 8, 9, 10 and 12 etc.

Factors and Multiples

Multiples

A number that is completely divisible by another number is a multiple of that number

Let’s find multiples of 3 using a number line.

Start from 0 and count on in 3 equal steps. You will find multiples of 3

3, 6, 9, 12, 15, 18, 21, 24, 27 and 30 are first ten multiples of 3.

Let’s find multiples of 4 using a number line

Recall the multiplication table of 4.

Factors and Multiples

Factors

Let’s arrange 8 balls in equal groups

There is more than one way to do this.

The number 8 can be divided by 1, 2, 4 and 8 without leaving a remainder.

1, 2, 4 and 8 are called factors of 8.

Factors of any numbers which divide the given number exactly. When we divide a number by its factors, the remainder is 0.

Start with the smallest number and Find all the numbers that divide 40 completely without leaving a remainder

Stop listing when numbers start to repeat.

The factors of 40 are 1, 2, 4, 5, 8, 10, 20 and 40.

Is 3 a factor of 25?

25 cannot be exactly divided by 3

 It leaves 1 as a remainder.

So, 3 is not a factor of 25

Factors and Multiples

Prime Factorization

We can express a number as a product of its prime factors.

Factor Tree Method

Example 1: Find prime factors of 27

Start with the smallest prime number that is a factor of 27.

27 is not divisible by 2. It is divisible by 3.

So, we divide 27 by 3.

Recall 3 × 9 = 27. Write 3 and 9 as two branches of 27,

3 is a prime number. So, we have one prime factor.

 9 is not a prime number. So, we further factorize it.

Recall 3 × 3 = 9. Write 3 and 3 as two branches of 9.

We get two more factors as 3 × 3

So, 27 = 3 × 3 × 3.

Writing a number as a product of its prime factors is called prime factorization

Division method

Example 1: Find prime factors of 18.

Start with the smallest prime number that divides 18 completely.

 9 is the smallest prime number that divides 18 without leaving a remainder.

 Recall 18 ÷ 2 =9

 So, we write 2 on the left side of 18 separating them with a line. 2 is a prime factor of 18.

 9 is not a prime number.

 So, we divide It again with the smallest prime number that divides it completely.

Recall 9 ÷ 3=3.

3 is a prime number.

We keep dividing until we get all prime numbers.

So, prime factors of 18 = 2 x 3 x 3

Factors and Multiples

Common multiples

Example 1 : Can you find common multiples of 2 and 3?

Let’s write first ten multiples of 2,

 2 4 6 8 10 12 14 16 18 20

 Let’s write first ten multiples of 3.

3 6 9 12 15 18 21 24 27 30

You can see that some numbers are multiples of both 2 and 3. We call them common multiples. Let’s circle common multiples of 2 and 3.

Common multiples of 2 and 3 are 6, 12 and 18.

6 Is the smallest number which is a common multiple of both 2 and 3.

It is called the lowest (or least) common multiple (LCM)

Least Common Multiple (LCM)

To find LCM, we fallow the following steps

Step 1 : Find multiples of all numbers

Step 2 : Circle the common Multiples

Step 3 : Find the lowest (or least) common multiple or LCM

Factors and Multiples

Finding LCM using prime factorization

We can also find LCM using prime factorization method.

 Example 1: Let’s find LCM of 6 and 8 using prime factorization method.

 Step 1 : Write prime factors of both numbers.

 Prime factors of 6 = 2 x 3

Prime factors of 8 = 2 x 2 x 2

Step 2 : Find the common factors.

 Common factors = 2

 Step 3: Write the Factors which are not common.

 Remaining factors = 3, 2, 2

Step 4: Multiply Factors from step 2 and step 3 to find LCM

LCM = 2 x 3 x 2 x 2 = 24.

Example 2: Finding LCM of 18 and 27

Let’s follow the steps we have learnt.

Prime factors of 18 = 2 × 3 × 3

 Prime factors of 27 = 3 × 3 × 3

Common factors = 3, 3 (Write all occurrences)

 Remaing factors = 2, 3 LCM = 3 × 3 × 2 × 3 = 54

Factors and Multiples

Common factors and Highest Common Factor (HCF)

Example 1: Can you find common factors of 12 and 18?

 Let’s write factors of 12.

 1 2 3 4 6 12

Let’s write factors of 18.

 1 2 3 6 9 18

You can see that some numbers are factors of both 12 and 18. We call them common factors. Let’s circle common factors of 12 and 18.

Common factors = 1, 2, 3, 6.

6 is the greatest number which is the common factor of both 12 and 18.

it is called the highest common factor (HCF)

To find HCF, we follow the following steps.

Step 1 : Find factors of all numbers.

 Step 2 : Circle the common factors.

Step 3 : Find the highest common factor or HCF

Factors and Multiples

Finding HCF using prime factorization

We can also find HCF using prime factorization method.

Example 1: Let’s find HCF of 6 and 24 using prime factorization method.

Step 1 : Find prime factors of both numbers.

Prime factors of 6 : 2 × 3

Prime factors of 24 : 2 × 3 × 2 × 2

Step 2 : Find the common factors Common factors are 2 and 3

Step 3 : Multiply the common factors to find HCF.

HCF = 2 × 3 = 6

Example 1: Find HCF of 30 and 45.

Let’s follow the steps we have learnt

Prime factors of 30 = 2 × 3 × 5

 Prime factors of 45 = 3 × 3 × 5

 Common factors = 3,5

 HCF = 3 × 5 = 15