Introduction
Is it possible to divide the integer number 72 by 6 evenly?
Yes, 72÷6=12
Is it possible to divide the integer number 72 by 7 without remainder?
No, 72÷7=10 r.2
Is it possible to determine all the numbers into which the integer number 72 can be divided?
Yes, this will be discussed in this article.
Are these numbers special?
These numbers are called factors.
Definition of factors
The factor of a number is the number that divides evenly the given number. Usually when we talk about number factors, we only talk about positive whole factors.
For example, the factors of 10 are 1, 2, 5 and 10. If 10 is divided by any of the numbers 1, 2, 5, 10, the answer will be a whole number:
10÷1=10
10÷2=5
10÷5=2
10÷10=1
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 exactly 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). To write consecutive prime numbers, starting with the smallest, you can use the algorithm sieve of Eratosthenes known before our era. Consider how this algorithm works on the following example.
EXAMPLE: Write down all prime numbers to 72.
SOLUTION: Write a list of whole numbers starting from the smallest prime number 2 to 72.
Circle the number 2 and cross out all numbers that are divisible by 2.
Circle the next number 3 and cross out all numbers that are divisible by 3.
Circle the next number 5 and cross out all numbers that are divisible by 5.
Circle the next number 7 and cross out all numbers that are divisible by 2.
And so on, circle the next numbers and cross out all numbers evenly divisible by this number. All circled numbers are prime numbers from 2 to 72.
A number that has more than two factors is called a composite number.
Ways of finding factors of a number
There are three methods of finding factors of a number:
- division method;
- multiplication method;
- prime factorization method.
DIVISION METHOD: the division method is to find all divisors from 1 to the number itself that divide the number without remainder.
MULTIPLICATION METHOD: the multiplication method is to write the number as a product of two numbers in different possible ways.
PRIME FACTORIZATION METHOD: the prime factorization method is to express a composite number as the product of its prime factors.
Division method of finding the factors of 72
Since number 72 is not circled in the example above (algorithm sieve of Eratosthenes), number 72 is a composite number, so it has more than two factors.
We can use division to find all positive factors of 72 (start with 1, then check 2, 3, 4, 5, 6, 7, 8, 9, etc. up to 36 (number 36 is exactly half of 72) and the number 72 itself):
72÷1=72
72÷2=36
72÷3=24
72÷4=18
72÷5=14 r.2
72÷6=12
72÷7=14 r.2
72÷8=9
72÷9=8
72÷10=7 r.2
72÷11=6 r.6
72÷12=6
72÷13=5 r.7
72÷14=5 r.2
72÷15=4 r.12
72÷16=4 r.8
72÷17=4 r.5
72÷18=4
72÷19=3 r.15
72÷20=3 r.12
72÷21=3 r.9
72÷22=3 r.6
72÷23=3 r.3
72÷24=3
72÷25=2 r.22
72÷26=2 r.20
72÷27=2 r.18
72÷28=2 r.16
72÷29=2 r.14
72÷30=2 r.12
72÷31=2 r.10
72÷32=2 r.8
72÷33=2 r.6
72÷34=2 r.4
72÷35=2 r.2
72÷36=2
72÷72=1
Therefore, 72 has a total of 12 positive factors: 1, 2, 3, 4, 6, 8, 9 12, 18, 24, 36, and 72.
Multiplication method of finding the factors of 72
Represent the number 72 as a product of two numbers in all different possible ways:
72= 1 × 72
72= 2 × 36
72= 3 × 24
72= 4 × 18
72= 6 × 12
72= 8 × 9
All the numbers that are used in these products are the factors of 72. Thus, the positive factors of 72 are 1, 2, 3, 4, 6, 8, 9 12, 18, 24, 36, and 72.
Prime factorization method of finding the factors of 72
Prime factorization is a way of expressing a number as a product of its prime factors. For example,
72=6×6×2
is not the prime factorization of 72 as number 6 is not a prime number.
The product
72=2×2×2×3×3
is the prime factorization of 72 as numbers 2 and 3 are prime numbers.
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 72 using the division method.
SOLUTION:
STEP 1: The smallest possible prime number that evenly divides 72 is number 2. Divide 72 by 2:
72÷2=36
STEP 2: The smallest possible prime number that evenly divides the quotient 36 is number 2. Divide 36 by 2:
36÷2=18
STEP 3: The smallest possible prime number that evenly divides the quotient 18 is number 2. Divide 18 by 2:
18÷2=9
STEP 4: The smallest possible prime number that evenly divides the quotient 9 is number 3. Divide 9 by 3:
9÷3=3
STEP 5: The smallest possible prime number that evenly divides the quotient 3 is number 3. Divide 3 by 3:
3÷3=1
LAST STEP: Multiply all the prime factors that are the divisors:
72=2×2×2×3×3
You can rewrite this prime factorization in the exponent form:
72=2232
REMARK: Usually this process is not described for so long, but depicted in the form of a column as shown below.
| 2 | 72 |
| 2 | 36 |
| 2 | 18 |
| 3 | 9 |
| 3 | 3 |
| 1 |
Prime factorization of 72=22233
=2332
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 a factor tree for the number 72.
SOLUTION: From the multiplication table we know that 72=8×9. Therefore,
Each of these numbers we can represent as products of prime numbers:
8=2×2×2
9=3×3
So, the final factor tree is
and the prime factorization of 72 is 2×2×2×3×3 or 2332.
and see that that the prime factorizations for all factor trees are the same. 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 next steps:
STEP 1: Draw a factor tree starting from arbitrary numbers with product equal to the given number but finishing with only prime numbers.
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 72 is 2332. Write down all positive factors of 72 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):
20=1, 21=2, 22=4, 23=8
- 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):
30=1, 31=3, 32=9
- and then all possible products of powers of both multipliers
2131=6, 2132=18,
2231=12, 2232=36,
2331=24, 2332=72
Therefore, the list of all 12 positive factors of 72 written in ascending order is
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Note that only 2 and 3 are prime factors of 72.
Finding the number of positive factors
Using the prime factorization of a number, we can find the number of factors of this number. To do this, complete the following steps:
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 72.
SOLUTION: From the previous topic, the prime factorization of the number 72 in the exponent form is 2332.
Add 1 to each exponent:
3+1=4
2+1=3
and multiply the obtained sums:
4×3=12
Therefore, the number of factors of 72 is 12.
Positive factor pairs of 72
A factor pair of a number is a set of two factors, which, when multiplied, give this number as a product. For example, factors 5 and 8 form a factor pair of 40 because
40=5×8
EXAMPLE: List all factor pairs of 72.
SOLUTION: Start with 1. Since 72=1×72, put 1 at the beginning of the list and 72 at the end of the list. Factors 1 and 72 form the first factor pair of 72.
| 1 | 72 |
Now, try 2. Since 72=2×36, put 2 at the beginning of the list (after 1) and 36 at the end of the list (before 72). Factors 2 and 36 form the second factor pair of 72.
| 1 | 2 | 36 | 72 |
Then, try 3. Since 72=3×24, put 3 at the beginning of the list (after 2) and 24 at the end of the list (before 36). Factors 3 and 24 form the third factor pair of 72.
| 1 | 2 | 3 | 24 | 36 | 72 |
Continue with 4. Since 72=4×18, put 4 at the beginning of the list (after 3) and 18 at the end of the list (before 24). Factors 4 and 18 form the fourth factor pair of 72.
| 1 | 2 | 3 | 4 | 18 | 24 | 36 | 72 |
Check the next factor 6. Since 72=6×12, put 6 at the beginning of the list (after 4) and 12 at the end of the list (before 18). Factors 6 and 12 form the fifth factor pair of 72.
| 1 | 2 | 3 | 4 | 6 | 12 | 18 | 24 | 36 | 72 |
At last, 72=8×9. Two middle factors are 8 and 9. These two factors form the last factor pair of 72.
| 1 | 2 | 3 | 4 | 6 | 8 | 9 | 12 | 18 | 24 | 36 | 72 |
There are no more whole numbers between 8 and 9 so we are done!
We can graphically represent positive factor pairs as shown below.
Negative factors of 72
From the previous topic you can see that there are 6 factor pairs of 72. All these pairs consist of positive whole numbers.
If we note that the product of two negative numbers gives a positive number, then the same pairs of negative numbers will also be factor pairs.
Therefore, negative factor pairs are
-1 and -72
-2 and -36
-3 and -24
-4 and -18
-6 and -12
-8 and -9
and list of negative factor pairs is
| -1 | -2 | -3 | -4 | -6 | -8 | -9 | -12 | -18 | -24 | -36 | -72 |
Here two numbers colored in one color form a negative factor pair.
Quiz
- What is the product of all prime factors of 72?
SOLUTION: Number 72 has only two prime factors: 2 and 3. The product of these two numbers is 2×3=6.
ANSWER: 6
- What is the mean of all composite factors of 72?
SOLUTION: Number 72 has 12 positive factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72. Factors 2 and 3 are prime factors, factor 1 is neither prime nor composite number, so there are 9 composite factors of 72:
4, 6, 8, 9, 12, 18, 24, 36, 72
The mean number is the sum of all numbers divided by the number of numbers. Therefore, the mean of all composite factors of 72 is
4+6+8+9+12+18+24+ 36+729=1899=21
ANSWER: 21
- What is the sum of all positive factors of 72?
SOLUTION: Number 72 has 12 positive factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72. The sum of these factors 1+2+3+4+6+8+9+12+18+24+36+72=195.
ANSWER: 195
- Which of the following statements is true?
a) The sum of all factors of 72 is 0.
b) The product of all factors of 72 is 0.
c) Number 72 has 24 factors.
d) Number 72 has 6 factor pairs.
SOLUTION: Number 72 has 24 factors: 12 positive and 12 negative. So, statement c) is true.
Positive factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Negative factors of 72: -1, -2, -3, -4, -6, -8, -9, -12, -18, -24, -36, -72.
The sum of all these factors is 0 because the sum of positive and corresponding negative factor is 0 and the sum of twelve zeros is zero too. Therefore, statement a) is true.
Statement b) is false. The product of numbers is 0 when one of these numbers is 0. The number 0 is not a divisor of the number 72, so the product of the factors of the number 72 can not be equal to 0.
Statement d) is false too because the number 72 has 6 positive factor pairs and 6 negative factor pairs, 12 factor pairs in total.
ANSWER: a) True
b) False
c) True
d) False
- Why numbers -12 and -4 do not form a factor pair of 72?
SOLUTION: A factor pair of a number is a set of two numbers whose product is equal to this number.
Since -12-4=48≠72, numbers -12 and -4 do not form a factor pair of 72 although both of these numbers are divisors of 72.
Conclusions
- Most often we are interested only in the positive factors of a number.
- Sometimes negative factors have some mathematical interest.
- We are never talking about fractional factors of a number.
- The number of factors of a number is the same as the number of prime factors of this number only if given number is a prime number.
- The number of factors of a composite number is always greater than 2.
- All natural numbers are the product of at least one factor pair.
- Number 1 is a factor of every natural number.
- When dividing, a divisor and a quotient always form a factor pair.
- A non-square number has an even number of factors.
- A square number has an odd number of factors.
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