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;
- Find the common prime factors
- 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:
- 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.
- 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
- 5 + 1 + 0 + 6 + 6 = 18
18 is divisible by 9.
Thus 51 066 is divisible by 9.
- 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:
- 225
- 194
- 342
- 484
1 thought on “Classwork Series and Exercises {Mathematics- JSS1}: Whole Numbers and Factors”
Find the factors of the following numbers?