What is the prime factorization of a number?
what is the prime factorization of 72??????
Wow
expressing a given number in factored form so that each of the factors are prime numbers.
e.g. 60 = 2x2x3x5
Factorization
It might come as a surprise to you but the breaking up of a number into its prime factors has been the unending pursuit of mathematicians for centuries. Lets take the layman's approach.
* Factoring or factorization is the process of breaking down a composite number into its prime factors and their respective exponents.
* Every composite number can be factored into a unique set of prime factors N = (p^a)(q^b)(r^c).....
* An integer p > 1 is called a prime number, or a prime, when its only divisors are +/- 1 and +/- p.
* Any number m > 1 that is not a prime is called a composite and results from multiplying primes together.
* The number 1 is the only number that is a factor of all other numbers.
*--Two integers, a and b, are said to be relatively prime when their greatest common divisor is 1.
* The factors, or divisors, of a number N are any numbers that evenly divide it. For example, the factors/divisors of
28 are 1, 2, 4, 7, 14 and 28. While all of these are also evenly divisible by their negative counterparts, it is universally accepted that the positive factors/divisors are what is meant.,
* The proper factors/divisors of a number N are any numbers that evenly divide N other than N itself. For example, the proper factors/divisors of 12 are considered to be 1, 2, 3, 4 and 6. This is the generally accepted definition of proper factors/divisors.
There exists some disagreement as to whether the number 1 should be considered a proper factor. Some mathematicians believe that the proper factors of a number N are any numbers that evenly divide N other than 1 and N. For example, under this definition, the proper factors of 12 are 2, 3, 4 and 6.
* The total number of factors/divisors of a number N, less the number itself, are often referred to as the aliquot parts, or aliquot divisors, of the number.
* Determining primes - The Sieve of Eratosthenes
Write down the numbers from 1 to 100, 1000, etc.
Strike out every second number starting from 2, i.e., 4, 6, 8, etc.
Starting from the first remaining number, 3, cross out every third number, i.e., 3, 6, 9, 12, etc.
Starting from the first remaining number, 5, cross out every fifth number, i.e., 5, 10, 15, 20, etc.
Starting from the first remaining number, 7, cross out every seventh number, i.e., 7, 14, 21, 28, etc.
Continue the process until you have reached 100, 1000, etc.
The numbers remaining are the primes between 1 and 100, 1000,etc.
* Determining the prime factors of the number N:
Starting with the first prime 2, divide N by 2 as many times as possible. When you can no longer divide the resulting numbers by 2, try dividing by the next prime, 3. When you can no longer divide the resulting number by 3, try dividing by the next prime, 5. Continue this process until you reach 1 as the final division. Each of the primes that you successfully divided into the succession of numbers are the prime factors and the number of times you
were able to divide them is the exponent of each prime. If you successfully divide N by 2 3 times, the first prime factor of N is 2^3. If you successfully divide N by 3 2 times, the next prime factor is 3^2. If 5 does not evenly divide into N/[2^3(3^2)], then 7 is not a prime factor. Continuing this process until you reach 1 will produce all the prime factors of N.
Example: Find the prime factors of N = 72.
72/2 = 36, 36/2 = 18. 18/2 = 9, 9/2 = ng making 2^3 the first prime factor.
9/3 = 3, 3/3 = 1 making 3^2 the second prime factor.
Therefore, the prime factorization of 72 is 2^3, 3^2.
Try N = 3960.
3960/2 = 1980. 1980/2 = 990. 990/2 = 495. 495/2 = ng making 2^3 the first factor.
495/3 = 165. 165/3 = 55. 55/3 = ng making 3^2 the second factor.
55/5 = 11. 11/5 = ng making 5^1 the third factor.
11/11 = 1 making 11^1 the final factor.
Therefore, the prime factorization of 3960 are 2^3, 3^2, 5^1, 11^1.
A more graphic, or pictorial, way to derive, and visualize, the prime factors is through a device called a factor tree. Taking the number 3960 again:
..........................................3960
/ \
/ \
/ \
60 66
/ \ / \
6 10 6 11
/ \ / \ / \ \
2 3 2 5 2 3 11
We again derive the factors of 2-2-2-3-3-5-11 or 2^3( 3^2)5^1(11^1). The resemblance to a tree only evolves if you start out with the two factors of the number closest to sqrt(N).
Another way of graphically representing the factorization is through the factor ladder.
3960
/ \
I----------->2 1980
I / \
2^3 2 990
I / \
I-------------------->2 495
/ \
I------------------------>3 165
3^2 / \
I---------------------------->3 55
/ \
5^15-------------------------------->5 11
/ \
11^1------------------------------------>11 1
* Determining the total number of factors or divisors of a number N:
Having the prime factors and their exponents, the next goal is usually to determine the total number of factors/divisors of the number N. The total number of factors or divisors of a number N is simply the product of each individual prime factor with every other prime factor in all possible ways. To first determine exactly how many factors/divisors there are without doing the actual multiplcation, let the number N be written as N =
p1^a1(p2^a2)..............pr^ar where the pi's are the various different prime factors and ai the number of times pi occurs in the prime factorization. The total number of divisors of the number N is then derived from
d(N) = (a1 + 1)(a2 + 1).........(ar + 1)
Example: The number 60 = 2x2x3x5 or 2^2x3^1x5^1 with exponents of 2, 1, and 1 which results in the total number of factors/divisors being
d(N) = (2 + 1)(1 + 1)(1 + 1) = 12
namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 totaling 12 in all.
* How would you determine a number N having d(N) total factors/divisors?
This is most easily learned by an example:
Find the smallest number having 14 factors/divisors. The only way we can break down 14 is 14 = 7 x 2, and subtracting unity from each we derive 6 and 1 as exponents to apply to any primes we wish. To derive the smallest number with 14 factors/divisors, we must obviously apply our exponents to the smallest primes, the highest exponents applied to the lowest primes, namely 2 and 3 which results in N = 2^6 x 3^1 = 192.
Recognize that any number of the form p^6(q), where p and q are primes greater than unity, also has exactly 14 divisors.
Find the smallest number with 8 factors. 4x2 results from exponents of 3 and 1 which leads to 2^3x3^1 = 24 and 24 has factors of 1, 2, 3, 4, 6, 8, 12 and 24.
The first question you might have is "What is a prime factor?
Even more basic, "What is a factor?"
A factor is a number that evenly divides another given number.
Therefore, a prime factor is a prime number that evenly divides another given number.
You might now ask yourself "What is a prime number?"
........................................."What makes them unique?
What is the difference between them and other numbers?
How many prime numbers are there?
How frequently do they appear?
Lets examine a couple of simple numbers, say 48 and 17.
What numbers evenly divide into 48? I find that 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 do for 10 in all.
What numbers evenly divide into 17? I find that only 1 and 17 do.
How strange!
How about the numbers 72 and 29?
What numbers evenly divide into 72? I find that 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72 do so for 12 in all.
What numbers evenly divide into 29? I find that only 1 and 29 do.
Even more strange.
If you go through this exercise with a variety of numbers, you will soon discover many more examples of numbers that have many factors/divisors and other numbers that have only 2 factors/divisors.
You might have noticed that every number has 1 as a factor and every number has itself as a factor. They are sometimes referred to as given factors or fundamental factors.
By default, we might then say that the factors other than 1 and the number itself are more significant, more worthy of exploring, or possibly more interesting.
The factors/divisors that fall between 1 and the number itself are referred to as the proper factors, to isolate them from the fundamental factors.
It is worth noting that the number 1 is even more unique in that it has but one factor, which just so happens to be itself. The number 1 stands alone as the only number with but one factor/divisor.
History records that our mathematical forefathers decided that the infinite array of numbers having only 2 factors be defined as the class of numbers called prime numbers, perhaps because they represented that unique set of numbers having the least number of factors/divisors (excluding 1), all others having 3 or more factors/divisors.
They also decided to define the set of all other numbers having 3 or more factors as composite numbers as they are derived from the multiplication of prime numbers.
With that as a background, the following material presents you with many more interesting facts and properties of prime and composite numbers. I hope you find it informative and interesting.
9x8=72
The prime factorization of a number is the expression of that number as a product of its prime factors. In other words, it involves finding the prime numbers that, when multiplied together, equal the original number.
To determine the prime factorization of a number, you can follow these steps:
1. Start by dividing the number by the smallest prime number, which is 2. If the number is divisible by 2, divide it by 2 repeatedly until it is no longer divisible by 2.
2. Move on to the next prime number, which is 3. Divide the remaining number by 3, repeatedly dividing it by 3 until it is no longer divisible.
3. Continue this process with the next prime numbers: 5, 7, 11, and so on. Each time you find a prime number that divides evenly into the remaining number, divide it completely.
4. Keep repeating these steps until the remaining number becomes 1. At this point, you have found all the prime factors of the original number.
5. Write down the prime factors you found. The prime factorization is the product of these prime factors.
Here's an example: Let's say we want to find the prime factorization of the number 24.
Start by dividing 24 by 2, which gives 12. Divide 12 by 2 again, resulting in 6. Now, 6 is not divisible by 2 anymore, so move on to the next prime number, 3. Divide 6 by 3, resulting in 2.
At this point, 2 is also a prime number, so we stop dividing. The prime factors we found are 2, 2, and 3. Thus, the prime factorization of 24 is 2 x 2 x 3, or simply 2^3 x 3.