Prime Factorization with Exponents

๐Ÿ†Practice decomposition into prime numbers

What does it mean?

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 88 can be written as 232^3
The number 11761176 can be also written as 22ร—3ร—72 2^2\times3\times7^2

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Test yourself on decomposition into prime numbers!

einstein

Write all the factors of the following number: \( 6 \)

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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 22, if the number is not a multiple of 22 we will go to the next prime number which is 33, and so on. We will recurse that the first prime numbers are 2,3,5,7,11,13,17,19,2, 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 11 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.

image decomposition of numbers


For example

Let us factorize the number 6464 as prime factors

Let's ask: Can 6464 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
Let's ask: Can 3232 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
Let's ask: Can 1616 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
Let's ask: Can 88 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
Let's ask: Can 44 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
Let's ask: Can 22 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
We arrive at 11. 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=2ร—2ร—2ร—2ร—2ร—2=26 64=2\times2\times2\times2\times2\times2=2^6


Now let's look at a more advanced example of factoring natural numbers as a product of powers within an exercise:

676ร—8โˆ’21612ร—928= \frac{6^{76}\times8^{-2}}{16^{12}\times9^{28}}=

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:

ืคื™ืจื•ืง ื”ืื™ื‘ืจ 6

Let's ask: Can 55 be divided, leaving no remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
Let's ask: Can 33 be divided, without remainder, by 22, the smallest prime number?
The answer is no.
Let us ask: Can 33 be divided, without remainder, by the next prime number, i.e. 33?
The answer is yes.
Let us add to our illustration the 33 and also the result.

We arrive at the number 11, 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=2ร—3 6=2\times3


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Let's write it down in the exercise

(2ร—3)76ร—8โˆ’21612ร—928= \frac{(2\times3)^{76}\times8^{-2}}{16^{12}\times9^{28}}=

ื”ืื 8 ืžืชื—ืœืง ืœืœื ืฉืืจื™ืช ื‘ืžืกืคืจ ื‘ืžืกืคืจ ื”ืจืืฉื•ื ื™ ื”ืงื˜ืŸ ื‘ื™ื•ืชืจ- 2

Let's ask: Can 88 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
Let's ask: Can 44 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
Let's ask: Can 22 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
We arrive at 11, 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 8=2^3

Now we will update our exercise:

(2ร—3)76ร—(23)โˆ’21612ร—928= \frac{(2\times3)^{76}\times(2^3)^{-2}}{16^{12}\times9^{28}}=

ื”ืื 16 ืžืชื—ืœืง ืœืœื ืฉืืจื™ืช ื‘ืžืกืคืจ ื”ืจืืฉื•ื ื™ ื”ืงื˜ืŸ ื‘ื™ื•ืชืจ- 2

Let's ask: Can 1616 be divided, without leaving remainder, by 22, the smallest prime number?
The answer is yes.

Let's add to our illustration the 22 and also the result.

Let's ask: Can 88 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
Let's ask: Can 44 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
Let's ask: Can 22 be divided, without remainder, by 22, the smallest prime number?
The answer is yes.
Let's add to our illustration the 22 and also the result.
We arrive at 11. 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 16=2^4

We will now update our exercise:

(2ร—3)76ร—(23)โˆ’2(24)12ร—928 \frac{(2\times3)^{76}\times(2^3)^{-2}}{(2^4)^{12}\times9^{28}}

Let's continue with the factorization of the. 99:

ื”ืื 9 ืžืชื—ืœืง ืœืœื ืฉืืจื™ืช ื‘ืžืกืคืจ ื”ืจืืฉื•ื ื™ ื”ื‘ื - 3

Let's ask: Can the 99 be divided, without leaving remainder, by 22, the smallest prime number?
The answer is no.
Let's continue with the prime number that comes after 22
Let's ask: Can 99 be divided, without remainder, by the next prime number, that is 33?
The answer is yes.
Let's add to our illustration the 33 and also the result.

Let's ask: Can 33 be divided, without remainder, by 22, the smallest prime number?
The answer is no.
Let's continue with the prime number that comes after 22
Let's ask: Can 33 be divided, without remainder, by the next prime number, that is 33?
The answer is yes.
Let's add to our illustration the 33 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 9=3^2

We will now update our exercise:

(2ร—3)76ร—(23)โˆ’2(24)12ร—(32)28 \frac{(2\times3)^{76}\times(2^3)^{-2}}{(2^4)^{12}\times(3^2)^{28}}

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:

276ร—376ร—2โˆ’6248ร—356= \frac{2^{76}\times3^{76}\times2^{-6}}{2^{48}\times3^{56}}=

Great!

Notice the negative power in the numerator with base 22?

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:

276ร—376248ร—356ร—26= \frac{2^{76}\times3^{76}}{2^{48}\times3^{56}\times2^6}=

Well, now we can apply in the denominator the law of the product of powers with equal bases and add the exponents of base 22.

We will obtain:

276ร—376254ร—356= \frac{2^{76}\times3^{76}}{2^{54}\times3^{56}}=

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:

222ร—320 2^{22}\times3^{20}

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 33 ways to write the value of the number 88.

Solution:

8=2ร—4 8=2\times4

8=23 8=2ยณ

Answer:

8=2ร—2ร—2 8=2\times2\times2


Do you know what the answer is?

Exercise 2

Find 33 ways to express the value of 6464 from the power.

Solution:

82=64 8ยฒ=64

43=64 4ยณ=64

Answer:

26=64 2^6=64


Exercise 3

How will you be able to express the number 31253125 using the power of 55?

55=3125 5^5=3125

Solution:

55=3125 5^5=3125

Answer:

55 5^5


Check your understanding

Exercise 4

Find another way to express this number with exponents

Solution:

93=729 9^3=729

Answer:

36 3^6


Exercise 5

What number with exponent 55 gives us the following number?

X5=59059 X^5=59059

Solution:

95=59059 9^5=59059

Answer:

9 9


Do you think you will be able to solve it?

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=2ร—1224=2\times12
  • 24=3ร—824=3\times8
  • 24=4ร—624=4\times6
  • 24=2ร—2ร—2ร—324=2\times2\times2\times3

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 1 .


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