1. Factors

2. What is a factor?
The concept of factor is simply the reverse of multiple. If A is a multiple of B, then B is a factor of A.
For example, 24 is a multiple of 6, so 6 is a factor of Colloquially, we say “6 goes into 24”.
The factors of 24 are all those natural numbers by which 24 can be divided exactly: 1, 2, 3, 4, 6, 8, 12 and 24.
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3. Recognizing quickly all the factors of a given number is, of course, very useful.
A set of 24 cans can be put into groups of 2, 3, 4, 6, 8 or 12, as shown in the rectangular arrays below:
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4. The Transitive Property
Similarly to “is a multiple of”, the mathematical relationship “is a factor of” also possesses the transitive property. For example, any factor of 12 must be a factor of 24, because 12 is a factor of 24.
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5. Highest Common Factor (HCF)
This is a similar idea to the lowest common multiple. If we list all the factors of two numbers, the two sets of numbers may have some numbers in common. (Since 1 is a factor of all numbers, they must at least have this number in common!) The largest of these common factors is called the highest common factor (HCF) or greatest common factor (GCF).
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6. Highest Common Factor (HCF)
For example, with 24 and 32 we have the following two sets of factors:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Factors of 32: 1, 2, 4, 8, 16, 32.
The factors in common are 1, 2, 4 and 8. So the highest common factor is 8.
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7. What is a Prime Number?
The strict mathematical definition is that any number that has precisely two factors, and no more than two, is called a prime number.
In practice, we think of a prime number as a number that cannot be divided exactly by any number apart from 1 and itself.
The prime numbers are 2, 3, 5, 7, 11, 13, …
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8. The Sieve of Eratosthenes
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9. Product of Prime Factors.
24 can be written as a product of factors in a number of different ways: 1 x 24, 2 x 12, 3 x 8, or 4 x 6.
As a product of Prime Factors, there is only one combination that will produce 24: 2 x 2 x 2 x 3.
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10. Using Prime Factors to find a HCF.
What is the HCF of 72 and 96?
72 = 2 x 2 x 2 x 3 x 3
96 = 2 x 2 x 2 x 2 x 2 x 3
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11. Using Prime Factors to find a HCF.
72 = 2 x 2 x 2 x 3 x 3
96 = 2 x 2 x 2 x 2 x 2 x 3
72 = 24 x 3; 96 = 24 x 4.
The HCF of 72 & 96 = 24
24 = 2 x 2 x 2 x 3
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12. Another shortcut to HCF
What is the HCF of 75 and 125?
Any factor of 75 and 125 must also be a factor of (125 – 75), i.e. 50.
Let’s start with 50, test if it is the required number, then progress down to the next highest factor of 50 (25, 10, 5, 2, 1) until we arrive at the required answer.
50 won’t “go into” either 75 or 125, but 25 will, therefore 25 is our required answer.
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13. What is the HCF of 48 and 80?
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14. What is the HCF of 48 and 80? 48 = 2 x 2 x 2 x 2 x 3
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15. The year 2003 was a prime year
The year 2003 was a prime year. When next will the year be a prime number?
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16. The year 2003 was a prime year
The year 2003 was a prime year. When next will the year be a prime number?
Not 2004 (even number)
Not 2005 (divisible by 5)
Not 2006 (even number)
Not 2007 ( = 9, therefore 2007 is divisible by both 3 and 9)
Not 2008 (even number)
Not 2009 (= 7 x 287)
Not 2010 (divisible by 2, 5 and 10)
2011 looks a likely answer!
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17. Which of the following is equivalent to 144?
22 x 34
23 x 32
23 x 33
24 x 32
24 x 33
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18. Which of the following is equivalent to 144?-
24 x ( 16 x 9 = 144 )
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19. Which of the following numbers is not a prime number?5 7 11 13 17 19 23 29 31 37 41 43 47 49 53 59
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20. Which of the following numbers is not a prime number?5 7 11 13 17 19 23 29 31 37 41 43 47 49 53 59
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21. What is the most number of rows in which 120 chairs (45 red and 75 blue) can be arranged so that each row has the same numbers of each colour?
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22. Note that 15 is the highest common factor of 45 and 75.
What is the most number of rows in which 120 chairs (45 red and 75 blue) can be arranged so that each row has the same numbers of each colour?
15 rows, each of 3 red and 5 blue, gives = 120 chairs in total.
Note that 15 is the highest common factor of 45 and 75.
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23. References
Mathematics Explained for primary teachers (third edition, 2006) by Derek Haylock.
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