Factors, Prime factors (revision)

40 ÷ 8 = 5 and 40 ÷ 5 = 8

8 and 5 divide into 40 without remainder.

8 and 5 are factors of 40.

A prime number has only two factors, itself and 1, but 1 is not a prime number.

2, 3, 5, 7, 11, 13, … are prime numbers.

Common factors

The number 12, 21 and 33 are all divisible by 3. We say that 3 is a common factor 0f 12, 21 33.

There may be more than one common factor of a set of numbers. For example, both 2 and 7 are common factors of 28, 42 and 70. Since 2 and 7 are common factors and are both prime numbers, then 14 (= 2 x 7) must also be a common factors of the set of numbers.

1 is a common factor of all numbers.

Highest Common Factor (HCF)

2, 7 and 14 are common factors of 28, 42 and 70; 14 is the greatest of three common factors. We say that 14 is the highest common factor of 28, 42 and 70.

To find the HCF of a set of numbers:

Express the number as a product of prime factors;

1. Find the common prime factors
2. Multiply the current prime factor together to give the HCF.

Example

Find the HCF of 18, 24 and 42.

18 = 2 x 3 x 3

24 = 2 x 2 x 2 3

42 = 2 x 3 x 7

The common prime factors are 2 and 3.

The HCF = 2 x 3 = 6.

Find the HCF of 216 and 288

2 | 216                              2 | 288

2 | 108                              2 | 144

2 | 54                                2 | 72

3 | 27                                2 | 36

3 | 9                                  2 | 18

3 | 3                                  3 |   9

……                                3 |    3

0      1                              ……..

0      1

In index notation

216 = 23 x 33

288 = 25 x 32

23 is the lowest power of two contained in the two numbers. Thus the HCF contains 23.

32 is the lowest power of 3 contained in the tow numbers. The HCF contains 32.

216 = (23 x 33) x 3

288 = (22 x 33) x 22

The HCF = 22 x 33 = 8 x 6 = 72

Rules of divisibility

Table 1.2 gives some rules for divisors of whole numbers.

 Any whole number is exactly divisible by … 2 if its last digit is even or 0 3 if the sum of its digit is divisible by 3 4 if its last two digits form a number divisible by 4 5 if its last digit is five or zero 6if its last digit is even and the sum of its digits is divisible by 3 8 if its lat three digits forma number divisible by 8 9if the sum of its digit is divisible by 9 10 if its last digit is 0

Table

There is no easy rule for division by 7.

Notice the following:

1. If a number m is divisible by another number n, m is also divisible by the factors of n. For example, a number divisible by 8 is also divisible by 2 and 4.
2. If a number is divisible by two or more numbers, it is also divisible by the LCM of these numbers. For example, a number divisible by both 6 and 9 is also divisible by 18, 18 is the LCM of 6 and 9.

Example

Test the following numbers to see which are exactly divisible by 9. a. 51 066 b. 9 039

Solution

1. 5 + 1 + 0 + 6 + 6 = 18

18 is divisible by 9.

Thus 51 066 is divisible by 9.

1. 9 + 0 + 3 + 9 = 21

21 is not divisible by 9.

Thus 9 039 is not divisible by 9.

Squares and square roots

Square roots

72 = 7 x 7 = 49.

In words ‘ the square of 7 is 49’. We can turn this statement round and say. ‘the square root of 49 is 7’.

In symbols, √49 = 7. The symbol √ means the square root of .

To find the square root of a number, first find its factors.

Example

Find √11 025.

Method: Try the prime numbers 2, 3, 5. 7, …

Working:

 3 11 025 3 3 675 5 1 225 5 245 7 49 7 7 1

11 025 = 32 x 52 x 72

= (3 x 5 x7) x (3 x 5 x 7)

= 105 x 105

Thus √11 025 = 105

It is not always necessary to write a number in its prime factors.

Example

√6 400

6400 = 64 x 100

= 82 x 102

Thus √6 400 = 8 x 10 = 80

The rules for divisibility can be useful when finding square root.

EXERCISES

Lets see how much you’ve learnt, attach the following answers to the comment below:

Find by factors the square roots of the following:

1. 225
2. 194
3. 342
4. 484