Any natural number that is not prime can be factorized as a product of prime numbers. This process is known as breaking down numbers into prime factors.
From time to time, to solve a certain exercise in a simpler way, we will have to dfactorize the numbers we are given, and rewrite them as products of prime numbers. This factorization gives the possibility to use the numbers in a more comfortable way, for example, by taking advantage of the properties of powers or exponent laws.
For example: The number 8 can be written as 23 The number 1176 can be also written as 22ร3ร72
Write all the factors of the following number: \( 6 \)
Incorrect
Correct Answer:
\( 2,3 \)
Practice more now
How is it done?
Let's write the number we want to factorize and draw a line next to it. We will ask ourselves what is the smallest prime number by which we can divide without having a remainder (zero remainder).
We will always start with 2, if the number is not a multiple of 2 we will go to the next prime number which is 3, and so on. We will recurse that the first prime numbers are 2,3,5,7,11,13,17,19, etc. If the number can be divided without remainder we will write the prime number in front of our original number and, below our number, we will write down the quotient (result of the division).
We will continue factoring until we get to the number 1 in the left column, this number can no longer be broken down. The number we wanted to factorize is equal to the product of all the numbers we wrote down in the right hand column.
For example
Let us factorize the number 64 as prime factors
Let's ask: Can 64 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. Let's ask: Can 32 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. Let's ask: Can 16 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. Let's ask: Can 8 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. Let's ask: Can 4 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. Let's ask: Can 2 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. We arrive at 1. We have finished decomposing. Now let's multiply all the factors we wrote down in the column on the right side, we will get: 64=2ร2ร2ร2ร2ร2=26
Now let's look at a more advanced example of factoring natural numbers as a product of powers within an exercise:
1612ร928676ร8โ2โ=
Don't be scared of the high exponents or that there is no equality of base.
That is exactly why we have learned to factor, that is, to decompose any natural number as a product of powers.
Let's go term by term and factor as we have learned.
After each factorization we will write down the exercise again so we don't get confused.
Let's start with:
Let's ask: Can 5 be divided, leaving no remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. Let's ask: Can 3 be divided, without remainder, by 2, the smallest prime number? The answer is no. Let us ask: Can 3 be divided, without remainder, by the next prime number, i.e. 3? The answer is yes. Let us add to our illustration the 3 and also the result.
We arrive at the number 1, we have finished factoring. Now let's multiply all the factors that we wrote down in the column on the right side and we will get:
6=2ร3
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Test your knowledge
Question 1
Write all the factors of the following number: \( 8 \)
Incorrect
Correct Answer:
\( 2,2,2 \)
Question 2
Write all the factors of the following number: \( 7 \)
Incorrect
Correct Answer:
No prime factors
Question 3
Write all the factors of the following number: \( 4 \)
Incorrect
Correct Answer:
\( 2,2 \)
Let's write it down in the exercise
1612ร928(2ร3)76ร8โ2โ=
Let's ask: Can 8 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. Let's ask: Can 4 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. Let's ask: Can 2 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. We arrive at 1, we have finished decomposing. Now let's multiply all the factors we wrote down in the column on the right side and we will get:
8=23
Now we will update our exercise:
1612ร928(2ร3)76ร(23)โ2โ=
Let's ask: Can 16 be divided, without leaving remainder, by 2, the smallest prime number? The answer is yes.
Let's add to our illustration the 2 and also the result.
Let's ask: Can 8 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. Let's ask: Can 4 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. Let's ask: Can 2 be divided, without remainder, by 2, the smallest prime number? The answer is yes. Let's add to our illustration the 2 and also the result. We arrive at 1. We have finished decomposing. Now let's multiply all the factors we wrote down in the column on the right side and we will get:
16=24
We will now update our exercise:
(24)12ร928(2ร3)76ร(23)โ2โ
Let's continue with the factorization of the. 9:
Let's ask: Can the 9 be divided, without leaving remainder, by 2, the smallest prime number? The answer is no. Let's continue with the prime number that comes after 2 Let's ask: Can 9 be divided, without remainder, by the next prime number, that is 3? The answer is yes. Let's add to our illustration the 3 and also the result.
Let's ask: Can 3 be divided, without remainder, by 2, the smallest prime number? The answer is no. Let's continue with the prime number that comes after 2 Let's ask: Can 3 be divided, without remainder, by the next prime number, that is 3? The answer is yes. Let's add to our illustration the 3 and also the result. Now let's multiply all the factors that we wrote down in the column on the right side and we will get:
9=32
We will now update our exercise:
(24)12ร(32)28(2ร3)76ร(23)โ2โ
Very good! We have finished factorizing all the natural numbers as a product of powers of prime numbers.
Now we will proceed according to the properties of powers.
Let's start with the first term in the numerator, in this case we will have to apply the power property of multiplication.
Then we will continue with the other terms where the power property will be appropriate.
Let's apply the properties, we will obtain:
248ร356276ร376ร2โ6โ=
Great!
Notice the negative power in the numerator with base 2?
Perfect, since this is our next step: eliminate the negative exponent. Clearly we will do this by placing the base in the denominator. We will obtain:
248ร356ร26276ร376โ=
Well, now we can apply in the denominator the law of the product of powers with equal bases and add the exponents of base 2.
We will obtain:
254ร356276ร376โ=
Great! Our next step will be, no doubt, to use the quotient property of powers with equal base: Subtract the exponent of the numerator minus the exponent of the numerator where there is equality of bases.
This step will relieve us of the nuisance of the fraction and we will be left with a much simpler expression with reduced exponents:
We will do this and obtain:
222ร320
Notice how factorizing numbers into prime factors in conjunction with the properties of exponents help to solve exercises in a very simple way.
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Exercises on Prime Factorization
Exercise 1
Find 3 ways to write the value of the number 8.
Solution:
8=2ร4
8=23
Answer:
8=2ร2ร2
Do you know what the answer is?
Question 1
Write all the factors of the following number: \( 5 \)
Incorrect
Correct Answer:
No prime factors
Question 2
Write all the factors of the following number: \( 9 \)
Incorrect
Correct Answer:
\( 3,3 \)
Question 3
Write all the factors of the following number: \( 12 \)
Incorrect
Correct Answer:
\( 3,2,2 \)
Exercise 2
Find 3 ways to express the value of 64 from the power.
Solution:
82=64
43=64
Answer:
26=64
Exercise 3
How will you be able to express the number 3125 using the power of 5?
55=3125
Solution:
55=3125
Answer:
55
Check your understanding
Question 1
Write all the factors of the following number: \( 14 \)
Incorrect
Correct Answer:
\( 2,7 \)
Question 2
Write all the factors of the following number: \( 16 \)
Incorrect
Correct Answer:
\( 2,2,2,2 \)
Question 3
Write all the factors of the following number: \( 13 \)
Incorrect
Correct Answer:
No prime factors
Exercise 4
Find another way to express this number with exponents
Solution:
93=729
Answer:
36
Exercise 5
What number with exponent 5 gives us the following number?
X5=59059
Solution:
95=59059
Answer:
9
Do you think you will be able to solve it?
Question 1
Write all the factors of the following number: \( 18 \)
Incorrect
Correct Answer:
\( 2,3,3 \)
Question 2
Write all the factors of the following number: \( 26 \)
Incorrect
Correct Answer:
\( 13,2 \)
Question 3
Write all the factors of the following number: \( 99 \)
Incorrect
Correct Answer:
\( 11,3,3 \)
Review questions
What does it mean to break down a number into factors?
It is when we represent a number as a product of others. For example:
24=2ร12
24=3ร8
24=4ร6
24=2ร2ร2ร3
How do we factorize?
To factorize we must make successive divisions until the quotient is one. For this we place our number and on the right side a vertical bar, we look for a divisor and place it on the right side, the quotient on the left side and repeat the process now looking for a divisor of the previously obtained quotient, until we get a one on the left side. Our initial number can be written as the product of all the divisors placed on the right side of the vertical line.
How we do prime factorization?
To break a number down into prime factors, we place the number and a vertical line to its right. We look for the smallest prime divisor of our number and we place it on the right side of the vertical line, the quotient we write it on the left side, just below the number we want to decompose. We repeat the process now looking for the smallest prime divisor of the quotient obtained previously and we continue this way until we obtain a quotient 1.
Test your knowledge
Question 1
Write all the factors of the following number: \( 31 \)
Incorrect
Correct Answer:
No prime factors
Question 2
Write all the factors of the following number: \( 500 \)
Incorrect
Correct Answer:
\( 5,5,5,2,2 \)
Question 3
Write all the factors of the following number: \( 6 \)
Incorrect
Correct Answer:
\( 2,3 \)
Examples with solutions for Prime Factorization with Exponents
Exercise #1
Write all the factors of the following number: 6
Video Solution
Step-by-Step Solution
To determine the factors of the number 6, we will follow these steps:
Step 1: Begin by checking each number starting from 1 up to 6 to see if it divides 6 evenly.
Step 2: Check 1. Since 1ร6=6, 1 is a factor.
Step 3: Check 2. Since 2ร3=6, 2 is a factor.
Step 4: Check 3. Since 3ร2=6, 3 is a factor.
Step 5: Check 4. Since 6 is not evenly divisible by 4, 4 is not a factor.
Step 6: Check 5. Since 6 is not evenly divisible by 5, 5 is not a factor.
Step 7: Finally, check 6. Since 6ร1=6, 6 is a factor.
All possible whole number products (pairs) that result in 6 are 1ร6, 2ร3, 3ร2, and 6ร1.
However, when checking for unique prime factors as a particular approach in factors identification, 6 breaks down into prime factors of 2 and 3.
Therefore, the primary distinct prime factors of 6 are 2 and 3.
This correlates with choice 3:
Choice 3: 2,3, which matches our factors.
Thus, the answer is correctly represented as the distinct prime factors 2,3 in the context of the problem requirements.
Answer
2,3
Exercise #2
Write all the factors of the following number: 8
Video Solution
Step-by-Step Solution
To find the factors of 8, we'll use prime factorization.
Step 1: Begin with the smallest prime number, 2.
Step 2: Divide 8 by 2, which gives 4. Since 4 is even, divide by 2 again.
Step 3: Divide 4 by 2, which gives 2. Divide 2 by 2 one more time.
Step 4: This yields 1, so we've fully factored the number.
Thus, the prime factorization of 8 is 2ร2ร2.
The factors of the number 8 are 2,2,2.
Therefore, the correct answer is choice 4: 2,2,2.
Answer
2,2,2
Exercise #3
Write all the factors of the following number: 7
Video Solution
Step-by-Step Solution
To determine all the factors of the number 7, we will examine which integers between 1 and 7 divide it exactly:
Check 1: Since 17โ=7, 1 is a factor.
Check 2: 27โ=3.5, which is not an integer, so 2 is not a factor.
Check 3: 37โโ2.333, which is not an integer, so 3 is not a factor.
Check 4: 47โ=1.75, which is not an integer, so 4 is not a factor.
Check 5: 57โ=1.4, which is not an integer, so 5 is not a factor.
Check 6: 67โโ1.167, which is not an integer, so 6 is not a factor.
Check 7: Since 77โ=1, 7 is a factor.
Therefore, the factors of 7 are 1 and 7.
These results correspond to choice 1: 1,7.
Answer
No prime factors
Exercise #4
Write all the factors of the following number: 4
Video Solution
Step-by-Step Solution
To solve this problem, we'll follow these steps:
Identify all the factors of the number 4.
Determine which of these factors are prime numbers.
Check the answer choices to find the one that corresponds to the prime factorization of 4.
Now, let's work through each step:
Step 1: To find the factors of 4, we consider pairs of numbers that multiply to 4, such as 1ร4 and 2ร2.
Step 2: Among these factors, identify the prime numbers. The number 2 is the only prime factor, and it needs to be listed twice since 2ร2=4.
Step 3: Looking at the answer choices, the choice that corresponds to the prime factorization of 4 is 2,2.
Therefore, the correct answer is 2,2.
Answer
2,2
Exercise #5
Write all the factors of the following number: 9
Video Solution
Step-by-Step Solution
To find all the factors of 9, we will determine the divisors of the number 9 by testing each integer from 1 up to 9.
Step 1: Test if 1 is a factor of 9. Since 9รท1=9, 1 is a factor.
Step 2: Test if 2 is a factor of 9. Since 9รท2=4.5 (not an integer), 2 is not a factor.
Step 3: Test if 3 is a factor of 9. Since 9รท3=3, 3 is a factor.
Step 4: Test if 4 is a factor of 9. Since 9รท4=2.25 (not an integer), 4 is not a factor.
Step 5: Test if 5 is a factor of 9. Since 9รท5=1.8 (not an integer), 5 is not a factor.
Step 6: Test if 6 is a factor of 9. Since 9รท6=1.5 (not an integer), 6 is not a factor.
Step 7: Test if 7 is a factor of 9. Since 9รท7โ1.2857 (not an integer), 7 is not a factor.
Step 8: Test if 8 is a factor of 9. Since 9รท8=1.125 (not an integer), 8 is not a factor.
Step 9: Test if 9 is a factor of 9. Since 9รท9=1, 9 is a factor.
The factors of 9 are 1, 3, and 9.
However, the problem might specifically be asking for the prime factorization where the number 9 decomposes into 3ร3.
Therefore, the correct answer which matches the provided choices is 3,3.