2 1 factors and prime numbers cimt org uk

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2 Factors MEP Y8 Practice Book A

2.1 Factors and Prime Numbers

A factor divides exactly into a number, leaving no remainder. For example, 13 is a factor of 26 because 26 ? 13 = 2 leaving no remainder. A prime number has only two factors, 1 and itself; this is how a prime number is defined. 5 is a prime number because it has only two factors, 1 and 5. 8 has factors 1, 2, 4 and 8, so it is not prime. 1 is not a prime number because it has only one factor, namely 1 itself.

Example 1

(a) List the factors of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. (b) Which of these numbers are prime numbers?

Solution

(a) This table lists the factors of these numbers:

Number

1 2 3 4 5 6 7 8 9 10

Factors

1 1, 2 1, 3 1, 2, 4 1, 5 1, 2, 3, 6 1, 7 1, 2, 4, 8 1, 3, 9 1, 2, 5, 10

(b) The numbers 2, 3, 5 and 7 have exactly two factors, and so only they are prime numbers.

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2.1

MEP Y8 Practice Book A

Example 2

List the prime factors of 24.

Solution

First list all the factors of 24, and they are: 1, 2, 3, 4, 6, 8, 12, 24

Now select from this list the numbers that are prime; these are 2 and 3, and so the prime factors of 24 are 2 and 3.

Example 3

Which of the following numbers are prime numbers: 18, 45, 79 and 90 ?

Solution

The factors of 18 are 1, 2, 3, 6, 9 and 18; 18 is not a prime number. The factors of 45 are 1, 3, 5, 9, 15 and 45; 45 is not a prime number. The factors of 79 are 1 and 79; 79 is a prime number The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90; 90 is not a prime number.

79 is the only prime number in the list.

Divisibility Test If a number is divisible by 2, it will end with 0, 2, 4, 6 or 8. If a number is divisible by 3, the sum of its digits will be a multiple of 3. If a number is divisible by 4, the last two digits will be a multiple of 4. If a number is divisible by 5, it will end in 0 or 5. If a number is divisible by 9, the sum of its digits will be a multiple of 9. If a number is divisible by 10, it will end in 0. Can you find tests for divisibility by other numbers?

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MEP Y8 Practice Book A

Exercises

1. (a) List all the factors of each of the following numbers: 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

(b) Which of these numbers are prime? 2. Explain why 99 is not a prime number. 3. Which of the following are prime numbers:

33, 35, 37, 39 ? 4. Find the prime factors of 72. 5. (a) Find the prime factors of 40.

(b) Find the prime factors of 70. (c) Which prime factors do 40 and 70 have in common? 6. Find the prime factors that 48 and 54 have in common. 7. A number has prime factors 2, 5 and 7. Which is the smallest number that has these prime factors? 8. The first 5 prime numbers are 2, 3, 5, 7 and 11. Which is the smallest number that has these prime factors? 9. Write down the first two prime numbers which are greater than 100. 10. Which is the first prime number that is greater than 200?

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MEP Y8 Practice Book A

2.2 Prime Factors

A factor tree may be used to help find the prime factors of a number.

Example 1

Draw a factor tree for the number 36.

Solution

Start with 36 and then:

split 36 into numbers 9 and 4 that multiply to give 36 as shown in the factor tree opposite;

repeat for the 9 and the 4, as shown on the factor tree.

The factor tree is now complete because the numbers at the ends of the branches are prime numbers; the prime numbers have been ringed.

36

9

4

3322

Another possible factor tree for 36 is shown here:

36

On the factor tree we only put a ring around the prime numbers.

12

3

62

Note that, at the end of the branches, both

the numbers 2 and 3 appear twice.

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The prime factors of 36 are 3, 2, 2 and 3.

In ascending order, the prime factors of 36 are 2, 2, 3, 3.

From the factor trees above it is possible to write:

36 = 2 ? 2 ? 3 ? 3 = 22 ? 32

When a number is written in this way, it is said to be written as the product of its prime factors.

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MEP Y8 Practice Book A

Example 2

Express each of the following numbers as the product of its prime factors:

(a) 102

(b) 60

Solution

(a) Start by creating a factor tree:

102

102 = 2 ? 3 ? 17

51

2

3 17

(b) Start by creating a factor tree: 60 = 5 ? 3 ? 2 ? 2

Put the prime numbers in ascending order: 60 = 2 ? 2 ? 3 ? 5

= 22 ? 3 ? 5

60

15

4

5322

Example 3

A number is expressed as the product of its prime factors as 23 ? 32 ? 5

What is the number?

Solution

23 ? 32 ? 5 = 2 ? 2 ? 2 ? 3 ? 3 ? 5 = 360

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