n3 multiples, factors and primes
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Learn about divisionTRANSCRIPT
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N3 Multiples, factors and primes
KS3 Mathematics
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Contents
N3 Multiples, factors and primes
A
A
A
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AN3.1 Divisibility
N3.5 HCF and LCM
N3.2 Multiples and factors
N3.3 Prime numbers
N3.4 Prime factor decomposition
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Divisibility by 2
Is 367 908 divisible by 2?
367 908 is divisible by 2 if 2 divides into it exactly without leaving any remainders.
We can tell if a number is divisible by 2 by looking at the last digit in the number.
Is 367 908
A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8.
A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8.
Yes, 367 908 is divisible by 2. It is an even number.
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Is 35 841 divisible by 3?
We can tell if a number is divisible by 3 by adding together its digits. This is called finding the digit sum.
A number is divisible by 3 if the digit sum is divisible by 3.
A number is divisible by 3 if the digit sum is divisible by 3.
The digit sum of 35 841 is 3 + 5 + 8 + 4 + 1 =
Divisibility by 3
21
21 is divisible by 3.
Yes, 35 841 is divisible by 3.
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Is 1 934 274 divisible by 4?
We can tell if a number is divisible by 4 by looking at the last two digits in the number.
Is 1 934 274
A number is divisible by 4 if the last two digits are divisible by 4.
A number is divisible by 4 if the last two digits are divisible by 4.
A two-digit number is divisible by 4:
Divisibility by 4
No, 1 934 274 is not divisible by 4.
if it ends in 2 or 6 and the first digit is odd.
if it ends in 0, 4 or 8 and the first digit is even
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Divisibility by 5
Is 231 030 divisible by 5?
We can tell if a number is divisible by 5 by looking at the last digit in the number.
Is 231 030
A number is divisible by 5 if the last digit is 0 or 5.A number is divisible by 5 if the last digit is 0 or 5.
Yes, 231 030 is divisible by 5.
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Is 53 702 divisible by 6?
Divisibility by 6
We can tell if a number is divisible by 6 by looking at its last digit and finding its digit sum.
A number is divisible by 6 if it is divisible by both 2 and 3.
A number is divisible by 6 if it is divisible by both 2 and 3.
The last digit in 53 702 is a 2 and so it is divisible by 2.
The digit sum of 53 702 is 5 + 3 + 7 + 0 + 2 = 17
53 703 is divisible by 6 only if it is divisible by both 2 and 3.
17 is not divisible by 3 and so 53 702 is not divisible by 3.
No, 53 702 is not divisible by 6.
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Is 63 072 divisible by 8?
We can tell if a number is divisible by 8 by halving it and checking for divisibility by 4.
A number is divisible by 8 if half of it is divisible by 4.
A number is divisible by 8 if half of it is divisible by 4.
Divisibility by 8
Look at the last two digits.
Half of 63 072 is 31 536
36 is divisible by 4.
Yes, 63 072 is divisible by 8.
31 536
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Is 6 873 975 divisible by 9?
We can tell if a number is divisible by 9 by finding its digit sum.
A number is divisible by 9 if its digit sum is divisible by 9.
A number is divisible by 9 if its digit sum is divisible by 9.
Divisibility by 9
The digit sum of 6 873 975 is 6 + 8 + 7 + 3 + 9 + 7 + 5 = 45
45 is divisible by 9.
Yes, 6 873 975 is divisible by 9.
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Divisibility by 10
Is 6 311 289 divisible by 10?
We can tell if a number is divisible by 10 by looking at the last digit in the number.
Is 6 311 289
A number is divisible by 10 if the last digit is 0.A number is divisible by 10 if the last digit is 0.
No, 6 311 289 is not divisible by 10.
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Divisibility by larger numbers
Is 214 875 divisible by 15?
We test for divisibility by some larger numbers by using two simpler tests.
A number is divisible by 15 if it is divisible by both 3 and 5.
A number is divisible by 15 if it is divisible by both 3 and 5.
The digit sum of 214 875 is 2 + 1 + 4 + 8 + 7 + 5 = 27
27 is divisible by 3 and so 214 875 is divisible by 3.
The last digit in 214 875 is a 5 and so it is also divisible by 5.
Yes, 214 875 is divisible by 15.
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Divisibility by larger numbers
We can check for divisibility by other larger numbers by testing for divisibility by factors of the number.
For example:
A number is divisible by 12 if it is divisible by both 3 and 4.
A number is divisible by 15 if it is divisible by both 3 and 5.
A number is divisible by 18 if it is divisible by both 2 and 9.
A number is divisible by 20 if it is divisible by both 4 and 5.
A number is divisible by 24 if it is divisible by both 3 and 8.
A number is divisible by 30 if it is divisible by both 3 and 10.
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Using divisibility to find multiples
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Contents
N3 Multiples, factors and primes
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A
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N3.2 Multiples and factors
N3.5 HCF and LCM
N3.1 Divisibility
N3.3 Prime numbers
N3.4 Prime factor decomposition
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Multiples
A multiple of a number is found by multiplying the number by any whole number.
A multiple of a number is found by multiplying the number by any whole number.
What are the first six multiples of 4?
To find the first six multiples of 4 multiply 4 by 1, 2, 3, 4, 5 and 6 in turn to get:
4, 8, 12, 16, 20 and 24.
Any given number has infinitely many multiples.
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Multiples patterns on a hundred square
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Rectangular arrangements
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Finding factors
A factor is a whole number that divides exactly into a given number.
A factor is a whole number that divides exactly into a given number.
Factors come in pairs.
For example, what are the factors of 30?
1 and 30, 2 and 15, 3 and 10, 5 and 6.
So, in order, the factors of 30 are:
1, 2, 3, 5, 6, 10, 15 and 30.
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Factor finder
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Circle and square puzzle
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Contents
N3 Multiples, factors and primes
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A
A
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A
N3.3 Prime numbers
N3.5 HCF and LCM
N3.1 Divisibility
N3.2 Multiples and factors
N3.4 Prime factor decomposition
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Sieve of Eratosthenes
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Prime numbers
If a whole number has two, and only two, factors it is called a prime number.
If a whole number has two, and only two, factors it is called a prime number.
For example, the number 17 has only two factors, 1 and 17.
Therefore, 17 is a prime number.
The number 1 has only one factor, 1.
Therefore, 1 is not a prime number.
There is only one even prime number. What is it?
2 is the only even prime number.
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The first 10 prime numbers are:
Prime numbers
2 3 5 7 11 13 17 19 23 29
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Testing for prime numbers
We can check whether or not a number is prime by testing for divisibility by successive numbers.
Is 107 divisible by 2? The last digit is a 7 so, no.
Is 107 divisible by 3? The digit sum is 8 so, no.
We don’t need to check for divisibility by 4 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it.
Is 107 divisible by 5? The last digit is a 7 so, no.
We don’t need to check for divisibility by 6 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it.
Is 107 a prime number?
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Testing for prime numbers
Is 107 a prime number?
We can check whether or not a number is prime by testing for divisibility by successive numbers.
Is 107 divisible by 7? Dividing by 7 leaves a remainder so no.
We don’t need to check for divisibility by 8 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it.
We don’t need to check for divisibility by 9 because if 3 doesn’t divide into 107, then no multiple of 3 can divide into it.
We don’t need to check for divisibility by 10 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it.
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Testing for prime numbers
We can check whether or not a number is prime by testing for divisibility by successive prime numbers.
Why don’t we need to check for divisibility by 11?We don’t need to check for divisibility by 11 because we have found that no number below 10 divides into 107.
That means that any number that multiplied 11 would have to be bigger than 10.
Since, 10 × 11 is bigger than 107 we can stop here.
107 is a prime number.
Is 107 a prime number?
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Testing for prime numbers
When we are testing whether or not a number is prime, we only have to test for divisibility by prime numbers.
We don’t need to check for divisibility by any number bigger than the square root of the number.
A number is prime if no prime number less than the square root of the number divides into it.
A number is prime if no prime number less than the square root of the number divides into it.
Also, all prime numbers greater than 5 must end in a 1, 3, 7 or 9.
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An amazing fact
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Contents
N3 Multiples, factors and primes
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A
A
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A
N3.4 Prime factor decomposition
N3.5 HCF and LCM
N3.1 Divisibility
N3.2 Multiples and factors
N3.3 Prime numbers
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A prime factor is a factor that is also a prime number.A prime factor is a factor that is also a prime number.
What are the factors of 30?
The factors of 30 are:
1, 2, 3, 5, 6, 10, 15, 30.
The prime factors of 30 are 2, 3, and 5.
Prime factors
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2 × 3 × 5 = 30
2 × 2 × 2 × 7 = 56 This can be written as 23 × 7 = 56
3 × 3 × 11 = 99 This can be written as 32 × 11 = 99
Every whole number greater than 1 is either a prime number or can be written as a product of
two or more prime numbers.
Every whole number greater than 1 is either a prime number or can be written as a product of
two or more prime numbers.
Products of prime factors
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The prime factor decomposition
When we write a number as a product of prime factors it is called the prime factor decomposition.
When we write a number as a product of prime factors it is called the prime factor decomposition.
The prime factor decomposition of 100 is:
There are 2 methods of finding the prime factor decomposition of a number.
100 = 2 × 2 × 5 × 5
= 22 × 52
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36
4 9
2 2 3 3
36 = 2 × 2 × 3 × 3
= 22 × 32
Factor trees
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36
3 12
4 3
2 2
36 = 2 × 2 × 3 × 3
= 22 × 32
Factor trees
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2100
30 70
6 5
2 3
10 7
2 5
2100 = 2 × 2 × 3 × 5 × 5 × 7
= 22 × 3 × 52 × 7
Factor trees
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780
78 10
392
3 13
25
780 = 2 × 2 × 3 × 5 × 13
= 22 × 3 × 5 × 13
Factor trees
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962
482
242
122
62
33
1
2
2
2
2
2
3
96 = 2 × 2 × 2 × 2 × 2 × 3
= 25 × 3
Dividing by prime numbers
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3153
1053
355
77
1
3
3
5
7
315 = 3 × 3 × 5 × 7
= 32 × 5 × 7
Dividing by prime numbers
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7022
3513
1173
393
1313
1
2
3
3
3
13
702 = 2 × 3 × 3 × 3 × 13
= 2 × 33 × 13
Dividing by prime numbers
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Contents
N3 Multiples, factors and primes
A
A
A
A
A
N3.5 HCF and LCM
N3.1 Divisibility
N3.2 Multiples and factors
N3.3 Prime numbers
N3.4 Prime factor decomposition
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Multiples of 6
Multiples of 8
24 48 72 968 16 32 40 56 64 80 88 104 …
612
18
24
3036
42
48
5460
66
72
7884
90
96
102…
Common multiples
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Multiples on a hundred grid
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The lowest common multiple
The lowest common multiple (or LCM) of two numbers is the smallest number that is a multiple of both the numbers.The lowest common multiple (or LCM) of two numbers is the smallest number that is a multiple of both the numbers.
We can find this by writing down the first few multiples for both numbers until we find a number that is in both lists.
For example:
Multiples of 20 are : 20, 40, 60, 80, 100, 120, . . .
Multiples of 25 are : 25, 50, 75, 100, 125, . . .
The LCM of 20 and 25 is 100.
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The first ten multiples of 8 are:
8, 16, 24, 32, 40, 48, 56, 64, 72, 80.
The first ten multiples of 10 are:
10, 20, 30, 40, 50, 60, 70, 80, 90, 100.
The lowest common multiple (LCM) of 8 and 10 is 40.
What is the lowest common multiple (LCM) of 8 and 10?
The lowest common multiple
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We use the lowest common multiple when adding and subtracting fractions.
Add together 4
9
5
12and .
The LCM of 9 and 12 is 36.
+4
9
5
12=
36
× 4
× 4
16+
36
× 3
× 3
15=
31
36
The lowest common multiple
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Common factor diagram
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The highest common factor
The highest common factor (or HCF) of two numbers is the highest number that is a factor of both numbers.The highest common factor (or HCF) of two numbers is the highest number that is a factor of both numbers.
We can find the highest common factor of two numbers by writing down all their factors and finding the largest factor in both lists.
For example:
Factors of 36 are :
1, 2, 3, 4, 6, 9, 12, 18, 36.
Factors of 45 are :
1, 3, 5, 9, 15, 45.
The HCF of 36 and 45 is 9.
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What is the highest common factor (HCF) of 24 and 30?
The factors of 24 are:
1, 2, 3, 4, 6, 8, 12, 24.
The factors of 30 are:
1, 2, 3, 5, 6, 10, 15, 30.
The highest common factor (HCF) of 24 and 30 is 6.
The highest common factor
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We use the highest common factor when cancelling fractions.
Cancel the fraction .36
48
The HCF of 36 and 48 is 12, so we need to divide the numerator and the denominator by 12.
36
48=
÷12
3
÷12
4
The highest common factor
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We can use the prime factor decomposition to find the HCF and LCM of larger numbers.
Find the HCF and the LCM of 60 and 294.
602302153551
60 = 2 × 2 × 3 × 5
29421473497771
294 = 2 × 3 × 7 × 7
Using prime factors to find the HCF and LCM
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60 294
60 = 2 × 2 × 3 × 5
294 = 2 × 3 × 7 × 7
22
35
7
7
HCF of 60 and 294 = 2 × 3 = 6
LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 = 2940
Using prime factors to find the HCF and LCM
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Using prime factors to find the HCF and LCM