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Factors & multiples

Article title

Questioning is the foundation of all learning.
The first step in rejecting not knowing is to ask, why?
Sweetland

Introduction

Overview

 

This page has information to explore factors as properties of whole numbers 1-50 and 1-100. It includes concepts for factors and related dimensions used to focus attention during instruction. Also includes an instructional sequence, focus questions and problems along with suggestions, factor tables, and worksheets.

Facts, Concepts, and Generalizations related to factors and multiples

Number value and operations

Multiples
  1. Multiples can be generated by skip counting. However, ... see related concepts.
  2. Multiples are products of unitized groups ( a number of groups and units in the group).
  3. Multiplicative (2n) is different than additive (n+n). Skip counting is additive (0) (2) (2 + 2) (2 + 2 + 2) .
  4. Multiple is a number that results from multiplying a given number by an integer (e.g., multiples of 0, 12 are 12, 24, 36, 48...).
  5. When learners count groups and then announce the product you can conclude they are unitizing and using a multiplicative strategy.
  6. Properties of multiples.
    1. Infinite: Every number has an endless sequence of multiples (they go on forever).
    2. Divisibility: A number is a multiple of another if it can be divided by that number with no remainder.
    3. Zero is a multiple: Zero is a multiple of every number (5 * 0 = 0).
    4. Self-multiple: Every number is a multiple of itself ((7 * 1 = 7).
  7. Common Multiple: A number that is a multiple of two or more different numbers. For example, 12 is a common multiple of both 3 and 4.
  8. Least Common Multiple (LCM): The smallest shared multiple between two or more numbers. For example, the common multiples of 4 and 6 include 12, 24, and 36, making 12 the least common multiple
Factors
  1. Factor is a number that divides evenly into another number (e.g., factors of 12 are 1, 2, 3, 4, 6, 12). Factors are finite.
  2. Factors are numbers that are multiplied together and result in a product. a * b = c; a and b are factors and c is the product.
  3. Multiply a number by another number and the result will have as many factors as the product (multiplicative relationship) of the factors (4 has 3 factors (1, 2, 4); 3 has 2 factors (1, 3); 12 has 6 factors (1, 2, 3, 4, 6, 12); 3 factors * 2 factors = 6 factors).
Primes
  1. A prime number has only two factors - itself and one. One is not a prime, it has only one factor.
  2. Every number has a unique (one and only one) prime factorization (Fundamental Theorem of Arithmetic).
  3. Primes only have two factors so if you multiply it by another prime, that makes three factors (2*1) [2 primes] (3*1) [2 primes or one more - since 1 is in both].
  4. When multiplying primes the exponent is one less than the number of factors ( 32 ) is 9 which has three factors. (1, 3, 9)

Data analysis

  1. Ordering by value can help discover relationships.
  2. Putting data in a chart can help distinguish relationships.
  3. Ordering by equality can help distinguish relationships.
  4. Venn diagrams can show relationships.

Reasoning and proof

  • Finding a counter example will disprove a conjecture.
  • Disproving a conjecture can provide insight.
  • When explaining (justify) my ideas it is important to select and include accurate information that is necessary and sufficient.
  • It's helpful to see and hear other peoples' ideas to get different views and insights.
  • Systematically exploring all possibilities can be used as a proof.

Multiples

 

Exploring factors

Instructional suggestions

Pose a question about the number of factorsa particlular number or set of numbers have and invite learners to discover them. Have them make representations of different numbers and their factors, organize them in a list or table. Use the table as a bridge to discover relationships.

When learners discover numbers with the same factors, introduce common factors.Which will open the door to relationships of factors to different products and common factors for different numbers. This will allow learners more opportunities to understanding multiplicative relationships for factors instead of an additive relationship.

And an introduction to greatest common factors (GCF) and prime numbers.

Focus questions & problems

  • What is a factor?
  • How many factors does a number have?
  • What number from 1-50 has the most factors?
  • What number from 1-50 has the least number of factors?
  • How many factors for the number one? Two? ...
  • Is there a relationship for the number of factors?
  • Can we figure it out?
  • What happens when the numbers are doubled?
  • What if we sort numbers (1-50) by the number of factors for each number?
  • How can we use the relationships we found to fine more factors? or Fill in the table?
  • What number form 1-100 has the most factors?
  • What number form 1-100 has the least number of factors?

 

Worksheets:

Primes

Finding primes and using them to find the GCF & LCM

 

 

Proof using the Fundamental Theory of Arithmatic

The following is a discussion of a conjecture and proof using the fundamental theory of arithmetic (FTA).

Set up

While exploring multiples and common multiples a learner says:

You can figure out the next (second) common multiple by doubling the least common multiple (LCM).

Then added, "Does it always work?"

When prompted to try to find a counter example, and after awhile they replied.

So far I cannot find a non example of this conjecture.

Do you know of any two numbers where this conjecture does not work???

Let's review what you have so far:

Conjecture: The second common multiple is always a double of the least common multiple (LCM).

Specific examples generated by the student:

Take two numbers. Say 3 and 4.
Find the first common multiple by listing all the multiples for each:

  • Common multiples of three: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...
  • Common multiples of four: 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, ...

Therefore, the first common multiple (LCM) is 12, the second is 24, the third is 36, ...

Therefore, for this example the first common multiple (LCM) is 12 and the second is 24; and 24 is twice 12.

Hence, the question ...

Will the second common multiple (24 in this example) always be twice the first?

Teacher decision time:

  • Should I stay with generating multiples for different numbers, compare them, and let the student see how all the examples are multiples of two and hope the ah ha moment will be all second multiples are two times the first, therefore, double. or
  • Use prime factorizations to show how the second multiple has the same factors as the first multiple and two. Therefore, as a multiple of two, a double.

An added bonus with the second method is: it can provide an excellent opportunity to introduce the Fundamental Theorem of Mathematics (FTA) as a stronger proof and to provide a wider application for the understanding of multiples, factors, and their relationships.

One way to proceed:

We could find for each multiple in each of the sets of multiples (3 and 4) all the prime factors for each multiple.

My reasoning is that if I can find all the smallest factors that are only primes, then I may be able to show a pattern to generalize for every multiple.

So a list of all the smallest factors (primes) for the first few multiples of each multiple for: three and four.

0
3 - 3
6 - 2, 3
9 - 3, 3
12 - 2, 2, 3
15 - 3, 5
18 - 2, 3, 3,
21 - 3, 7
24 - 2, 2, 2, 3 ---> notice this is exactly the same as the prime factors of 12 * 2 or double. Try some more examples... Can make it infinite?
27 - 3, 3, 3
30 - 2, 3, 5
33 - 3, 11
36 - 2, 2, 3, 3 ---> notice this is exactly the same as the prime factors of 12 * 3 or triple.

0
4 - 2, 2
8 - 2, 2, 2
12 - 2, 2, 3 ---> notice....
16 - 2, 2, 2, 2
20 - 2, 2, 5
24 - 2, 2, 2, 3 ---> notice...
28 - 2, 2, 7
32 - 2, 2, 2, 2, 2
36 - 2, 2, 3, 3 ---> notice ...

NOW this is where the student might see the pattern and recognize the answer to his or her question.

If they do or don't either way, it is still necessary to complete the next step, and illustrate the Fundamental Theorem of Arithmetic.

To get to that discovery, look at the prime factors for the first 25 whole numbers?

0 - not a prime by definition
1 - not a prime by definition
2 - 2
3 - 3
4 - 2, 2
5 - 5
6 - 2, 3
7 - 7
8 - 2, 2, 2
9 - 3, 3
10 - 2, 5
11 - 11
12 - 2, 2, 3
13 - 13
14 - 2, 7
15 - 3, 5
16 - 2, 2, 2, 2
17 - 17
18 - 2, 3, 3
19 - 19
20 - 2, 2, 5
21 - 3, 7
22 - 2, 11
23 - 23
24 - 2, 2, 2, 3
25 - 5, 5

Okay now, which of the numbers have exactly the same factors?

NONE?

WOW! Every whole number is a product of a unique set of prime numbers. In other words a group of primes has only one product. AND every whole number only has one set of prime numbers that can be used to make that number.

Now. If you are all still with me, put these two ideas together and think of
12 as the group of primes 2, 2, 3.

In order to get the next multiple, of that number, it would have to include another prime factor; and the smallest prime is 2.

Oh, wow!

  • 12 = 2 * 2 * 3 and multiply it by the smallest prime
  • 2 * 2 * 3 * 2
  • and where was something like that seen before?
  • 24 = 2 * 2 * 2 * 3

Summary

Let's represent primes to 30 on a number line:

Prime factors for numbers to thirty and the fundamental theory of arithmetic (FTA)

 

Therefore, if every whole number can be expressed as a unique group of primes, then any whole number (which would include any LCM for any group of numbers) would have as its next multiple one more factor and the smallest factor that it could be is two. SOoo, anyone that wants to find a counterexample that doesn't fit this students conjecture, probably will have a long search.

YES! Wasn't that fun?

Factor worksheets

Factors for numbers 1 - 50

Number Factors Number of Factors
1 1*1 ---------> [1]  
2 1*2 ---------> [1, 2]  
3 1*3 ---------> [1, 3]  
4 1*4, 2*2 ---------> [1, 2, 4]  
5 1*5 ---------> [1, 5]  
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Factors for numbers 1 - 50 - sorted by least to most

One
Factor
***		 Two
Factors

 Three
Factors
√√√		 Four
Factors

 Five Factors
∆∆∆		 Six
Factors

 Seven
Factors
ΩΩΩ		 Eight
Factors

 Nine
Factors
∑∑∑		 Ten
Factors
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   

 

Factors for numbers 1 - 100

Number Factors Number of Factors
1 1*1 ---------> [1] 1 ***
2 1*2 ---------> [1, 2] 2
3 1*3 ---------> [1, 3] 2
4 1*4, 2*2 ---------> [1, 2, 4] 3 √√√
5 1*5 ---------> [1, 5] 2
6 1*6, 2*3 ---------> [1, 2, 3, 6] 4
7 1*7 ---------> [1, 7] 2
8 1*8, 2*4, ---------> [1, 2, 4, 8] 4
9 1*9, 3*3 ---------> [1, 3, 9] 3 √√√
10 1*10, 2*5 ---------> [1,2 5, 10] 4
11 1*11 ---------> [1, 11] 2
12 1*12, 2*6, 3*4 ---------> [1, 2, 3, 4, 6, 12] 6
13 1*13 ---------> [1, 2] 2
14 1*14, 2*7 ---------> [1, 2, 7, 14] 4
15 1*15, 3*5 ---------> [1, 3, 5, 15] 4
16 1*16, 2*8, 4*4 ---------> [1, 2, 4, 8, 16] 5 ∆∆∆
17 1*17 ---------> [1, 17] 2
18 1*18, 2*9, 3*6 ---------> [1, 2, 29, 18] 4
19 1*19 ---------> [1*19] 2
20 1*20, 2*10, 4*5 ---------> [1, 2, 4, 5, 10, 20] 6
21 1*21 ---------> [1, 3, 7, 21] 2
22 1*22, 2*11 ---------> [1, 2, 11, 11] 4
23 1*23 ---------> [1, 23] 2
24 1*24, 2*12, 3*8, 4*6 ---------> [1, 2, 3, 4, 6, 8, 12, 24] 8
25 1*25, 5*5 ---------> [1, 5, 25] 3 √√√
26 1*26, 2*13 ---------> [1, 2, 13, 26] 4
27 1*27, 3*9 ---------> [1, 3, 9, 27] 4
28 1*28, 2*14, 4*7 ---------> [1, 2, 4, 7, 14, 28] 6
29 1*29 ---------> [1, 29] 2
30 1*30, 2*15, 3*10, 5*6 ---------> [1, 2, 3, 5, 6, 10, 15, 30] 7 ΩΩΩ
31 1* 31 ---------> [1, 31] 2
32 1*32, 2*16, 4*8 ---------> [1, 2, 3, 4, 8, 16, 32] 7  ΩΩΩ
33 1*33, 3*11 ---------> [1, 3, 11, 33] 4
34 1* 34, 2*17 ---------> [1, 2, 17, 34] 4
35 1*35, 5*7 ---------> [1, 5, 7, 35] 4
36 1*36, 2*18, 3*12, 4*9, 6*6 ---------> [1, 2, 3, 4, 6, 9, 12, 18, 36] 9 ∑∑∑
37 1*37 ---------> [1, 37] 2
38 1*38, 219, ---------> [1, 2, 19, 38] 4
39 1*39, 3*13 ---------> [1, 3, 13, 39] 4
40 1*40, 2*20, 4*10, 5*8 ---------> [1, 2, 4, 5, 8, 10, 20, 40] 8
41 1*41 ---------> [1, 41] 2
42 1*42, 2* 21, 3*14, 6*7 ---------> [1, 2, 3, 6, 7, 14, 21, 42] 8
43 1*43 ---------> [1, 43] 2
44 1*44, 2*22, 4*11, ---------> [1, 2, 4, 11, 22, 44] 6
45 1*45, 3*15, 5*9, ---------> [1, 3, 5, 9, 15, 45] 6
46 1*46, 2*23 ---------> [1, 2, 23, 46] 4
47 1*47 ---------> [1, 47] 2
48 1*48, 2*24, 3*16, 4*12, 6*8 ---------> [1,2, 3, 4, 6, 8, 12, 16, 24, 48] 10
49 1*49, 7*7 ---------> [1, 7, 49] 3 √√√
50 1*50, 2*25, 5*10, ---------> [1, 2, 5, 10, 25, 50] 6
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Factors for numbers 1 - 100 - sorted by least to most

1
***		 2

 3
√√√		 4

 5
∆∆∆		 6

 7
ΩΩΩ		 8

 9
∑∑∑		 10

 11 12 13 14 15 16 17 18 19 20
1 2 4 6 16 12 30 24 36 48                    
  3 9 8   18 32 40                        
  5 25 10   20   42                        
  7 49 14   28                            
  11   15   44                            
  13   21   45                            
  17   22   50                            
  19   26                                
  23   27                                
  29   33                                
  31   34                                
  37   35                                
  41   38                                
  43   39                                
  47   46                                
                                       

 

Factors for the First 50 Numbers

Factors for numbers 1 to 50
Number Factors Number of Factors
1 1*1 ---------> [1] 1 ***
2 1*2 ---------> [1, 2] 2
3 1*3 ---------> [1, 3] 2
4 1*4, 2*2 ---------> [1, 2, 4] 3 √√√
5 1*5 ---------> [1, 5] 2
6 1*6, 2*3 ---------> [1, 2, 3, 6] 4
7 1*7 ---------> [1, 7] 2
8 1*8, 2*4, ---------> [1, 2, 4, 8] 4
9 1*9, 3*3 ---------> [1, 3, 9] 3 √√√
10 1*10, 2*5 ---------> [1,2 5, 10] 4
11 1*11 ---------> [1, 11] 2
12 1*12, 2*6, 3*4 ---------> [1, 2, 3, 4, 6, 12] 6
13 1*13 ---------> [1, 2] 2
14 1*14, 2*7 ---------> [1, 2, 7, 14] 4
15 1*15, 3*5 ---------> [1, 3, 5, 15] 4
16 1*16, 2*8, 4*4 ---------> [1, 2, 4, 8, 16] 5 ∆∆∆
17 1*17 ---------> [1, 17] 2
18 1*18, 2*9, 3*6 ---------> [1, 2, 29, 18] 4
19 1*19 ---------> [1*19] 2
20 1*20, 2*10, 4*5 ---------> [1, 2, 4, 5, 10, 20] 6
21 1*21 ---------> [1, 21] 2
22 1*22, 2*11 ---------> [1, 2, 11, 11] 4
23 1*23 ---------> [1, 23] 2
24 1*24, 2*12, 3*8, 4*6 ---------> [1, 2, 3, 4, 6, 8, 12, 24] 8
25 1*25, 5*5 ---------> [1, 5, 25] 3 √√√
26 1*26, 2*13 ---------> [1, 2, 13, 26] 4
27 1*27, 3*9 ---------> [1, 3, 9, 27] 4
28 1*28, 2*14, 4*7 ---------> [1, 2, 4, 7, 14, 28] 6
29 1*29 ---------> [1, 29] 2
30 1*30, 2*15, 3*10, 5*6 ---------> [1, 2, 3, 5, 6, 10, 15, 30] 7 ΩΩΩ
31 1* 31 ---------> [1, 31] 2
32 1*32, 2*16, 4*8 ---------> [1, 2, 3, 4, 8, 16, 32] 7  ΩΩΩ
33 1*33, 3*11 ---------> [1, 3, 11, 33] 4
34 1* 34, 2*17 ---------> [1, 2, 17, 34] 4
35 1*35, 5*7 ---------> [1, 5, 7, 35] 4
36 1*36, 2*18, 3*12, 4*9, 6*6 ---------> [1, 2, 3, 4, 6, 9, 12, 18, 36] 9 ∑∑∑
37 1*37 ---------> [1, 37] 2
38 1*38, 219, ---------> [1, 2, 19, 38] 4
39 1*39, 3*13 ---------> [1, 3, 13, 39] 4
40 1*40, 2*20, 4*10, 5*8 ---------> [1, 2, 4, 5, 8, 10, 20, 40] 8
41 1*41 ---------> [1, 41] 2
42 1*42, 2* 21, 3*14, 6*7 ---------> [1, 2, 3, 6, 7, 14, 21, 42] 8
43 1*43 ---------> [1, 43] 2
44 1*44, 2*22, 4*11, ---------> [1, 2, 4, 11, 22, 44] 6
45 1*45, 3*15, 5*9, ---------> [1, 3, 5, 9, 15, 45] 6
46 1*46, 2*23 ---------> [1, 2, 23, 46] 4
47 1*47 ---------> [1, 47] 2
48 1*48, 2*24, 3*16, 4*12, 6*8 ---------> [1,2, 3, 4, 6, 8, 12, 16, 24, 48] 10
49 1*49, 7*7 ---------> [1, 7, 49] 3 √√√
50 1*50, 2*25, 5*10, ---------> [1, 2, 5, 10, 25, 50] 6

 

 

 

 

 

 

 

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