Numbers can be placed into different sets according to their properties. For example, 12 is an even number because when it is divided by 2 there is no remainder. Some of the different types of sets of numbers are: odd, even, prime, composite, factors and multiples.
When an even number is divided by 2, it doesn't have a remainder.
Circle all the even numbers in the following list:
3 17 8 19 20 64 82 101
156 225 403 800 978
On the other hand, when a number is divided by 2 and a remainder is left, that number is odd. Write all the odd numbers from the list above.
_____, _____, _____, _____, _____, _____
Did you notice that all the:
even numbers end with either the digit 0, 2, 4, 6 or 8 in the ones place.
odd numbers end with either the digit 1, 3, 5, 7 or 9 in the ones place.
Factors and multiples are different things but they both involve multiplication:
Factors are the numbers we can multiply to get a number.
1 × 6 = 6, so 1 and 6 are factors of 6
2 × 3 = 6, so 2 and 3 are factors of 6
There are no other 2 numbers that we can multiply to get 6. So, the factors of 6 are {1, 2, 3, 6}. This fact can proven by examining the numbers 4 and 5 which are missing from the list.
There is no number that I can multiply 4 with to get 6, so 4 is not a factor of 6.
There is no number that I can multiply 5 with to get 6, so 5 is not a factor of 6.
A factor of a given number is never greater than that number. Therefore, there is no need to examine numbers such as 7, 8, 9,.....because they are greater than 6. As you can see, 6 has only 4 factors. Some numbers have just a few factors while others may have quite more than just a few. In any case though, the set of factors is always finite. That means the set does't go on forever.
List the factors of 10.
1 x 10 = 10
2 x 5 = 10
The factors of 10 are {1, 2, 5, 10}
List the factors of 12.
1 x 12 = 12
2 x 6 = 12
3 x 4 = 12
Note that because 2 x 6 and 6 x 2 are the same factors, we don't have to repeat them. The same goes for 3 x 4 and 4 x 3.
Therefore the factors of 12 are {1 ,2 ,3 ,4 6, 12}
FACTS: The number 1 is a factor of all numbers.
Every number is a factor of itself.
List all the factors of :
15
18
29
How many factors does 14 have?
Can you explain why 7 is not a factor of 12?
The highest common factor (HCF), also known as the greatest common factor (GCF), of 2 or more numbers is the largest number that is a factor of all the numbers concerned. Follow these steps to find the HCF:
list the factors of each number.
find the factors that are common to all sets. Common factors are the numbers that are a part of all the sets. The number 1 is always common so it isn't considered.
from the common factors, choose the one that is greatest.
Example #1, what is the HCF of 6 and 9?
The factors of 6 are: {1, 2, 3, 6}
The factors of 9 are: {1, 3, 9}
The common factor is 3
Therefore, the HCF is 3
Example #2, what is the HCF of 6, 12 and 18?
The factors of 6 are: {1, 2, 3, 6}
The factors of 12 are: {1, 2, 3, 4, 6, 12}
The factors of 9 are: {1, 2, 3, 6, 9, 18}
The common factors are 2, 3 and 6. The largest is 6.
The HCF is 6
Multiples are what we get after multiplying numbers.
1 × 6 = 6, so 6 is a multiple of 6
2 × 6 = 12, so 12 is a multiple of 6
3 x 6 = 18, so 18 is a multiple of 6
and so on. The multiples of 6 are {6, 12, 18, 24, 30, 36, .......}
20 is not a multiple of 6 because there is no number you can multiply 6 by to get 20. Likewise, 55 is not but 246 is because 41 x 6 = 246.
The given number is always a multiple of itself and the set of multiples of any number is infinite. That means the set will continue forever.
