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Factors & multiples

Common multiples, GCF, LCM, primes, & the FTA

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

Overview

 

This page explores properties of whole numbers: multiples, factors, primes, least common multiple (LCM), greatest common factor (GCF) and the fundamental theory of arithmetic (FTA). It includes an instructional sequences, focus questions and problems along with suggestions, procedures, and worksheets for introducing and exploring these concepts.

Activity sequence

  • Introduce multiples and how to generate them.
  • Common multiples and least common multiple (LCM)
  • Factors
  • Greates common factor (GCF)
  • Primes
  • Using primes to find LCM & GCF
  • Fundamental theory of arithmetic (FTA)

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).
    Example
    3 + 2 is a group of 3 plus a group of 2 or 5.
    (⚽⚽⚽ ) + (⚽⚽) =
    (⚽⚽⚽⚽⚽)
    3 * 2 is three groups of two or 6.
    (⚽⚽)+(⚽⚽)+ (⚽⚽) =
    (⚽⚽)
    (⚽⚽)
    (⚽⚽)
  4. Multiple is a number that results from multiplying a given number by an integer (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

Background

Multiples can be generated by skip counting. However, multiples are multiplicative (2n) while skip counting is additive (n+n); (0) (2) (2 + 2) (2 + 2 + 2). More on multiplication representations.

Multiples are products of unitized groups (a number of groups and units in the group).

3 * 2 is three groups of two where Each group is the unit.
(⚽⚽)+(⚽⚽)+ (⚽⚽) =
(⚽⚽)
(⚽⚽)
(⚽⚽)

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...).

When learners count groups and then announce the product you can conclude they are unitizing and using a multiplicative strategy.

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).

Finding multiples

To find multiples of numbers you can suggest to start with the number, write it down, then think of another group of that number and write the number for the two groups. Then continue for 3, 4, 5, ... groups of the number.
See - Blank Find the multiples worksheet

Find the multiples

Directions: Find the multiples for these numbers.

Multiples of two.

(2, 4, 6, 8, 10, 12, 14, 16, 18, 20 ...)

Multiples of three.

(3, 6, 9, 12, 15, 18 ...)

Multiples of four.

(4, 8, 12, 16, 20, ...)

Multiples of five.

(5, 10, 15, 20, ...)

While it's good to remember that zero is a multiple of all numbers, it would be problematic when you are looking for least common multiples, as it would always be zero.

However, saying the LCM of 4 and 6 is zero instead of 12 provides no useful information for real-world tasks using a least common multiple. Therefore, we define the LCM to exclude zero by defining the LCM as the smallest positive common multiple.

While this can be assumed in elementary mathematics, some advanced mathematics use zero as a multiple.

Common multiples

A common multiple is a positive whole number that is a multiple of two or more numbers.

For example, 12 is a common multiple of both 3 and 4.

Since multiples and common multiples are infinite, lets limit them to less than 31 as we explore common multiples for some numbers.

To find common multiples we first find the multiples of the numbers.

For three and four less than 31:

Multiples of three.

(3, 6, 9, 12, 15, 18, 21, 24, 27, 30)

Multiples of four.

(4, 8, 12, 16, 20, 24, 28)

We can see the numbers in both sets of multiples, that are the same are 12 and 24. ∴ the common multiples of 3 & 4 < 31 are 12 & 24.

We can illustrate this with a Venn diagram.

Venn diagram multiples 3 & 4
Common multiples of three and four

 

Summary

Least Common Multiple (LCM): The smallest shared multiple between two or more numbers.

For example,
The common multiples of three and four are 12, 24, 36, ...
With the least common multiple 12.

Find the common multiple for the set of numbers on this worksheet! and identify the LCM.

 

Find the multiples

Directions: Find the multiples for these numbers.

Multiples of two.

(2,

Multiples of three.

(3,

Multiples of four.

(4,

Multiples of five.

(5,

Multiples of

(

 

 

Find the common multiples

Directions: Select numbers, find the common multiples less than ____, complete a Venn diagram, and write the LCM.

Multiple bank starters ...

(2, 4, 6, 8, 10, 12, 14, 16, 18, 20

(3, 6, 9, 12, 15, 18

(4, 8, 12, 16, 20,

(5, 10, 15, 20,

Multiples Venn diagram

What is the least common multiple (LCM)

Multiples Venn diagram

What is the least common multiple (LCM)

 

 

Factors

Instructional suggestions and sequence

Background

  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).

Finding factors

Pose a question about the number of factors a particlular number or set of numbers have and invite learners to discover them.

Ask. How did you know or find them?

Use their answer to explore finding factors.

Possibly using responses to lead to examples like below.

Example for factors of six

Ask. How can we find the factors of six? accept all answers and record a procedure to generalize for finding factors.

Possible example:

Start by making an open set of parentheses as you ask the question.

(       )

What is the first and last factor? It is always one and the number.
So put them in the parentheses.

(1,       6)

Go to the next positive whole number - two and ask if it is a factor of the number (6)?
If it is, put it and its factor in the set. write 2 , 3 in parentheses.

If not ask if the next positive whole number is and so on till you meet a factor you have in the parentheses.

(1, 2,   3, 6)

Say. OH! look 2 and 6. We met in the middle so we have all the factors.

Let's check. Is four a factor of six? no

Is five? no

Then we have found all the factors of six.

For fun, let's try a larger number.

 

Example for factors of 36

Ask. How can we find the factors of thirty-six? build on previous example to generalize a procedure for finding factors.

Example:

Making an open set of parentheses as you ask the question.

(            )

Put the first and last factor in the parentheses.

(1,            36)

Ask. What do we do next? We go to the next positive whole number - two and ask if it is a factor of the number (36)?
If it is, put it and its factor in the set. write 2 , 18 in parentheses.

(1, 2,        18, 36)

Ask. What do we do next? Go to the next positive whole number -3 and ask if it is a factor of 36?
It is, so write it and its factor in the parentheses.

(1, 2, 3,       12, 18, 36)

Ask. What do we do next? Go to the next positive whole number - 4 and ask if it is a factor of 36? Yes 4 *9 is 36 and

(1, 2, 3, 4,     9, 12, 18, 36)

Ask. What do we do next? Go to the next positive whole number -6 and ask if it is a factor of 36?
It is, so write it and its factor in the parentheses.

(1, 2, 3, 4,  6   9, 12, 18, 36)

OH! look 6 and 6. We met in the middle so we have all the factors.

Wow! 36 has a nice set of factors.

 

Factors as multiplicative

Have them make representations of different numbers and their factors to demonstrate the multiplicative aspect of factors. Factor representation worksheet

Representation for factors of 12

 

Create a procedure to factor any number

After the learners become familiar with finding factors ask them if we can write a procedure together for factoring numbers.

Factorization heuristic:

  1. Make open parentheses with plenty of space between the open and closing parenthesis.
  2. Enter the smaller factor, one next to the left parenthesis or opening parenthesis
  3. Enter the larger factor next to the right parenthesis or closing parenthesis.
  4. You now have the smallest and largest factors.
  5. Ask if two is a factor. (Hint all even numbers have two as a factor.) If it is, write a comma next to one, and enter two (smaller factor of the next factor pair).
  6. Find the larger factor that when multiplied by two equals the number you are factoring.
  7. Write it to the left of the number being factored, next to the closing parenthesis and follow it with a comma.
  8. Check to see if three, is a possible factor,
  9. If it is, then add it to the right of two followed by a comma.
  10. Write its factor pair left of the last factor entered on the right side and place a comman after it.
  11. Continue increasing by one checking for the next factor and writing them in the list as you find them.
  12. When the value of the factors meets, you have found them all.

Congratulations!

That is a lot of words and not what most of us want to take class time writing it all out. How about abrief summary.

Summary.

To find all the factors of a number make a comma separated list from one to the number being factored. Trying all whole numbers one at a time until you meet.

To help with find factors check out Memorizing basic facts & algorithms - Development of multiplication and division facts for ideas on how recongnize multiples.

 

Moving on to mastery of factorization

Review what we know and make a list or table of factors to explore factors and use as a bridge to discover relationships.

Note

When learners discover numbers with the same factors, mention common factors.And when they discover ones with no factors, but one and itself, then mention primes. This will open the door to relationships of factors to different products and common factors for different numbers. And consider how to include opportunities to understand multiplicative relationships for factors instead of an additive relationship. This will set the stage to develop the greatest common factor (GCF) and prime factorization.

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 between numbers?
  • 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 find more factors? or Fill in the table?
  • What number from 1-100 has the most factors?
  • What number from 1-100 has the least number of factors?

To explore and begin to answer these questions we can use the following:

Worksheets:

As learners explore and find the factors for the numbers 1-50 or 1-100 the will be developing fluency. After which they will be ready to explore primes, primefactorization, and the fundamental theory of arithmetic (FTA).

 

Primes

Background

  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 and it in't a prime. (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)

Finding primes

Pose a question about a particlular number or set of numbers as being primes and invite learners to discover some.

Ask. How did you know or find them?

Use their answer to explore finding primes.

Possibly work their responses to lead to using factor trees for prime factorization.

Factor tree example

To use factor trees to find a prime factorization.

  1. Write the number.
  2. Is it a prime?
  3. If it is circle it and you are done.
  4. If it isn't, draw two lines below for its factors.
  5. Write the factors at the end point of each line.
  6. Continue the process until only primes are left.

 

Factor tree examples 2-10

 

Let's try one a bit more fun!

Like 60.

Prime factor tree for 60

 

The factors of 60 are (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) Wow! that's a lot!

If we pair them up and find the prime factors for each factor.

We see that the four primes are all used in each, only in different orders.

1, 60 - (1) (60 - 2, 2, 3, 5)

2, 30 - (2) (30 - 2, 3, 5)

3, 20 - (3) (20 - 2, 2, 5)

4, 15 - (4 - 2, 2) (15 - 3, 5)

5, 12 - (5) (12 - 2, 2, 3)

6, 10 - (6v- 2, 3) (10 - 2, 5)

 

Try 72!

Prime factor for 72

 

Finding primes and using them to find the GCF & LCM

 

Prime factor tree for 96 & 90

 

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

Factor representations

Representation for factors of 12

 

Complete the labels for the factors of 12 reprentation.

Make one of your own below.

 

 

 

 

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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100    

 

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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