**Introduction**

Imagine that you have a cuboid made of identical Lego bricks that you want to divide into equal cuboids. How many ways can you do this? Is it possible to mathematically describe such a procedure in order not to guess the solution? In which cases will there be a single result? If you are interested, this article will give you answers to your questions.

**Definition of factors of **45

The factor of a number is the number that divides evenly the given number. For example, if you divide 45 by 3, you get 15, so number 3 is a factor of 45 (3 divides 45 evenly). Moreover, number 15 is a factor of 45 too because 45 divided by 15 is 3 (15 divides 45 evenly).

**REMEMBER:** 45 divided by a factor of 45 will equal another factor of 45.

Factors could be positive or negative. In most cases, we talk about positive factors, because the problems associated with negative factors occur much less often and for them, you can draw similar conclusions as for positive factors. Try to think in which case there may be a need of negative factors?

**Prime and composite numbers**

The factors of a number are any numbers that divide into it exactly, including 1 and the number itself. Number 1 is always the smallest whole factor of the number, the number itself is always the greatest whole factor of the number. So, each number always has at least two factors.

If a number has only two factors, 1 and the number itself, this number is called a prime number. Number 1 is not a prime number. The smallest prime number is number 2 (is evenly divisible by 1 and by 2). A number that has more than two factors is called a composite number.

**EXAMPLE**: Determine whether following numbers are prime or composite.

a) 21;

b) 47;

c) 153.

**SOLUTION:** a) From the multiplication table we know that 21=3×7. In addition to the two trivial factors 1 and the number itself, the number 21 has two more factors 3 and 7. So, number 21 has more than 2 factors and is a composite number.

b) When we try to find the factor of a number, we divide this number by all consecutive whole numbers that do not exceed half of this number. Dividing number 47 by all natural numbers from 2 to 23, we can see that every time there is a non-zero remainder. Therefore, number 47 has only two factors, 1 and the number itself, so this number is a prime number.

c) Applying divisibility rule by 9, we can see that the sum of the digits 1+5+3=9 of number 153 is evenly divisible by 9, so number 153 has 9 as a factor. Thus, number 153 is a composite number as it has more than two factors.

**Ways of finding factors of a number**

Factors of any number n can be calculated by many methods:

- division method;
- multiplication method;
- prime factorization method.

DIVISION METHOD: the division method is to divide a given number n by all natural numbers from 1 to n. All numbers that give a zero-remainder divided by this number are factors of the given number n.

MULTIPLICATION METHOD: the multiplication method is to write the number n as a product of two numbers in different possible ways.

PRIME FACTORIZATION METHOD: the prime factorization method is to represent a composite number n as the product of its prime factors.

**Division method of finding the factors of **45

Number 45 ends with 5, so according to the divisibility rules, this number is evenly divisible by 5. Hence, number 45 has more than two factors and is a composite number.

Follow the steps to calculate the factors of 45:

STEP 1: Write down the number to be factored, i.e., 45.

STEP 2: Find the numbers which can evenly divide 45:

45÷1=45

45÷2=22 r.1

45÷3=15

45÷4=11 r.1

45÷5=9

45÷6=7 r.3

45÷7=6 r.3

45÷8=5 r.5

45÷9=5

45÷10=4 r.5

45÷11=4 r.1

45÷12=3 r.9

45÷13=3 r.6

45÷14=3 r.3

45÷15=3

45÷16=2 r.13

45÷17=2 r.11

45÷18=2 r.9

45÷19=2 r.7

45÷20=2 r.5

45÷21=2 r.3

45÷22=2 r.1

45÷23=1 r.22

45÷45=1

STEP 3: There are 6 numbers which do not leave a remainder: 1, 3, 5, 9, 15 and 45. These numbers are the factors of 45.

**Multiplication method of finding the factors of **45

Represent the number 45 as a product of two whole numbers in all different possible ways:

45= 1 × 45

45= 3 × 15

45= 5 × 9

All the numbers that are used in these products are the factors of 45. So, the positive factors of 45 are 1, 3, 5, 9, 15 and 45.

**Prime factorization method of finding the factors of **45

Prime factorization is a method of representing the given number as the product of prime numbers. For example,

45=5×9

is not the prime factorization of 45 as number 9 is not a prime number.

The product

45=3×3×5

is the prime factorization of 45 as numbers 3 and 5 are prime numbers.

If a number occurs more than once in prime factorization, it is usually expressed in exponential form to make it more compact. In our case,

45=32×5

There are two possible ways to express number as a product of prime factors:

- division method;
- factor tree method.

PRIME FACTORIZATION USING DIVISION METHOD

The division method can be used to find the prime factors of a number by dividing the number by prime numbers. To find the prime factors of a number using the division method complete the next steps:

STEP 1: Evenly divide the number by the smallest possible prime number.

STEP 2: Evenly divide the quotient of step 1 by the smallest possible prime number.

STEP 3: Repeat step 2, until the quotient becomes 1.

LAST STEP: Multiply all the prime factors that are the divisors.

**EXAMPLE:** Write the prime factorization of the number 45 using the division method.

**SOLUTION:**

STEP 1: Determine the smallest possible prime factor of the number 45 and divide 45 by this factor:

45÷3=15

STEP 2: Find the smallest possible prime factor of the quotient 15 and divide 15 by this factor:

15÷3=5

STEP 3: Determine the smallest possible prime factor of the quotient 5 and divide 5 by this factor:

5÷5=1

LAST STEP: Multiply all the prime factors:

45=3×3×5

Collapse the result in the exponent form:

45=32×5

**REMARK 1: **Very often we only write a column “prime factorization” in order not to describe the process for so long, but only to write down the quotients and factors.

3 | 45 |

3 | 15 |

5 | 5 |

1 |

Prime factorization of 45=3 x 3 x 5

=3^{2}×5

**REMARK 2: **Another quick way of finding a complete prime factorization of any given number is to use what is essentially “upside-down” division and dividing only by the smallest prime that can fit into each result. Let’s demonstrate this with the example of the number 45.

STEP 1: Divide the number 45 by the smallest possible prime factor. In this case, it’s 3.

STEP 2: The quotient is 15. Now divide 15 by the smallest possible prime factor. In this case, it’s again 3.

STEP 3: This time the quotient is 5. Now divide 5 by the smallest possible prime factor. In this case, it’s 5.

STEP 4: At last, the quotient is 1, so we finished the division. The prime factorization will be all the number on the “outside” (circled numbers) multiplied together.

So, the prime factorization of 45 is 3×3×5.

PRIME FACTORIZATION USING FACTOR TREE METHOD

A factor tree is a special diagram, where we draw the factors of a number, then the factors of those factors and so on until we get only prime factors.

**EXAMPLE**: Draw an arbitrary factor tree for the number 45.

**SOLUTION**: From the multiplication table we know that 45=5×9. Therefore,

One of these numbers we can represent as products of prime numbers:

9=3×3

So, the final factor tree is

and the prime factorization of 45 is 3×3×5 or 32×5.

**REMEMBER**: Determining prime factorization by the factor tree method, everyone can start with different pair of factors. For example, instead of 5 and 9 you can take 3 and 15.

The final factorization must be always the same as the prime factorization is **unique**. This is provided by the Fundamental Theorem of Arithmetic which states that

**every natural number greater than **1** can be written as a product of prime numbers, and that up to rearrangement of the factors, this product is unique.**

To find the prime factors of a number using the factor tree method complete the following steps:

STEP 1: Draw a factor tree starting with arbitrary factors which product equals to the given number but finishing with only prime factors.

STEP 2: Multiply all the prime factors that appear in this factor tree.

The method of finding the factors of a number using the prime factorization of the number is to multiply arbitrary combinations of prime factors.

**EXAMPLE**: The prime factorization of 45 is 3^{2}×5. Write down all positive factors of 18 using the prime factorization method.

**SOLUTION**: To write down all the factors and not lose any, use the following guidance of writing combinations:

- first, write down all possible powers of the first prime factor (starting with exponent of 0 and finishing with the maximum possible exponent defined in the prime factorization of the number):

3^{0}=1, 3^{1}=3, 3^{2}=9

- then write down all possible powers of the second prime factor (starting with exponent of 0 and finishing with the maximum possible exponent defined in the prime factorization of the number):

5^{0}=1, 5^{1}=5

- and then all possible products of powers of both multipliers

3^{1} x 5^{1}=15, 3^{2} x 5^{1}=45

Therefore, the list of all 6 positive factors of 45 written in ascending order is

1, 3, 5, 9, 15, 45

Note that only 3 and 5 are prime factors of 45.

**Finding the number of positive factors**

The following algorithm helps to determine the number of positive factors of the number 45:

STEP 1: Write the prime factorization of a number in the exponent form.

STEP 2: Add one to each of the exponents.

STEP 3: Multiply all obtained numbers. This product denotes the number of factors of a number.

**EXAMPLE**: Find the number of factors of 45.

**SOLUTION**: From the previous topic, the prime factorization of the number 45 in the exponent form is 3^{2} x 5.

Add 1 to each exponent:

2+1=3

1+1=2

and multiply the obtained sums:

3 x 2=6

Therefore, the number of factors of 45 is 6.

**Positive factor pairs of **45

A factor pair of a number is a pair of two factors, which, when multiplied, give this number as a product. For example, factors 7 and 6 form a factor pair of 42 because

42=7×6

**EXAMPLE:** List all positive factor pairs of 45.

**SOLUTION:** Start with 1. Since 45=1×45, put 1 at the beginning of the list and 45 at the end of the list. Factors 1 and 45 form the first factor pair of 45.

1 | 45 |

Now, try 3. Since 45=315, put 3 at the beginning of the list (after 1) and 15 at the end of the list (before 45). Factors 3 and 15 form the second factor pair of 45.

1 | 3 | 15 | 45 |

Then, try 5. Since 45=59, put 5 at the beginning of the list (after 3) and 9 at the end of the list (before 15). Factors 5 and 9 form the third factor pair of 45.

1 | 3 | 5 | 9 | 15 | 45 |

There are no more whole factors of 45 between 5 and 9 so we are done!

We can graphically represent positive factor pairs as shown below.

**Negative factor pairs of **45

The previous topic shows that there are 3 positive factor pairs of 45. If we take one such pair of positive factors, change them to the corresponding negative numbers and multiply these negative numbers, we get a negative factor pair too. For example, the pair (3, 15) is a positive factor pair, so the pair (-3, -15) is a negative factor pair.

Therefore, negative factor pairs are

-1 and -45

-3 and -15

-5 and -9

and list of negative factor pairs is

-1 | -3 | -5 | -9 | -15 | -45 |

Here two numbers colored in one color form one negative factor pair.

Also, we can graphically represent negative factor pairs in the same way as graphical representation of positive factor pairs.

**Quiz**

- Some friends plucked 45 oranges from a tree and distributed the fruit among themselves equally. If there the number of friends is one-digit number greater than 6, how many oranges did each friend get?

**SOLUTION**: Positive factor pairs of 45 are (1, 45), (3, 15) and (5, 9).

There is the only one one-digit factor greater than 1 – this is number 9.

So, the number of friends plucking oranges is 9 and each of them gets

45÷9=5 oranges

**ANSWER:** 5 oranges

- How many factors of 18 are also factors of 45?

**SOLUTION**: Number 18 has 6 positive factors:

1, 2, 3, 6, 9, 18

Number 18 has 6 positive factors:

1, 3, 5, 9, 15, 45

List of common factors:

1, 3, 9

But we count only positive factors. Numbers -1, -3 and -9 are common factors two. So, there are 6 common factors.

**ANSWER:** 6 factors (3 positive and 3 negative)

- Which of the following statements is true?

a) Each factor of 15 is a factor of 45.

b) Each factor of 45 is a factor of 15.

c) Each factor of 5a is a factor of 45a, where a is a natural number.

**SOLUTION**: a) Number 15 has factors 1, 3, 5, 15. Number 45 has factors 1, 3, 5, 9, 15, 45. As you can see all factors of 15 are factors of 45, so statement a) is true.

b) Number 45 has factors 1, 3, 5, 9, 15, 45. Number 15 has factors 1, 3, 5, 15. As you can see not all factors of 45 are factors of 15, so statement b) is false.

c) Suppose number n is a factor of 5a. This means there exists such number x that

n×x=5a

Now, represent number 45 in the following way:

45a=9×5a

Substitute n×x instead of 5a:

45a=9×(n×x)

45a=9x×n

Therefore, number n evenly divides number 45a and is its factor by definition of a factor.

**ANSWER**: a) True

b) False

c) True

- The product of the three whole numbers is 45. What could be the sum of these numbers? Which sum is the greatest? Which sum is the smallest?

**SOLUTION:** From this article we know that the number 45 has the following factors:

1, 3, 5, 9, 15, 45

Fill in the following table:

Product | 1^{st} number | 2^{nd} number | 3^{rd} number | Sum |

45 | 1 | 1 | 45 | 1+1+45=47 |

45 | 1 | 3 | 15 | 1+3+15=19 |

45 | 1 | 5 | 9 | 1+5+9=15 |

45 | 3 | 3 | 5 | 3+3+5=11 |

The greatest sum is 45, the smallest sum is 11.

**ANSWER:** Possible sums: 11, 15, 19, 47. The greatest sum is 45, the smallest sum is 11.

- CHALLENGE QUESTION: Number A is a factor of 45 and number B is a multiple of 5. When A=B?

**SOLUTION:** Factors of 45:

1, 3, 5, 9, 15, 45

Multiples of 5:

5, 10, 15, 20, 25, 30, 35, 40, 45, …

(we are not interested in multiples greater than 45 because number A cannot be greater than 45).

List all common numbers in both sets:

5, 15, 45

Hence, there are three possibilities when A=B:

A=B=5

A=B=15

A=B=45

**ANSWER:** Three possibilities: A=B=5, A=B=15, A=B=45

**Conclusions**

- Number 45 is composite number (it has 12 factors).
- Number 45 has 6 positive factors: 1, 3, 5, 9, 15, 45 and 6 negative factors: -1, -3, -5, -9, -15, -45.
- 6 positive factors of 45 form 3 factor pairs as well as 6 negative factors.
- The number of factors can be found using exponent from of prime factorization.
- There are only 3 different factor trees of number 45.

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