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? ______________________________________________
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.
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
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
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
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
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
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
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