Factoring into prime numbers allows you to decompose a number into its basic elements. If you don't like working with large numbers, like 5,733, you can learn to represent them in a simpler way, for example: 3 x 3 x 7 x 7 x 13. This type of process is indispensable in cryptography or in the techniques used to guarantee information security. If you're not ready to develop your own secure email system yet, start using prime factorization to simplify fractions.
Steps
Part 1 of 2: Factoring into Prime Factors
Step 1. Learn factoring
It is a process of "breaking down" a number into smaller parts; these parts (or factors) generate the starting number when multiplied with each other.
For example, to decompose the number 18, you can write 1 x 18, 2 x 9, or 3 x 6
Step 2. Review the prime numbers
A number is called prime when it is divisible only by 1 and by itself; for example, the number 5 is the product of 5 and 1, you cannot break it down further. The purpose of prime factorization is to factor each value down until you get a sequence of prime numbers; this process is very useful when dealing with fractions to simplify their comparison and use in equations.
Step 3. Start with a number
Choose one that is not prime and greater than 3. If you use a prime number, there is no procedure to go through, as it is not decomposable.
Example: The prime factorization of 24 is proposed below
Step 4. Divide the starting value into two numbers
Find two that, when multiplied together, produce the starting number. You can use any pair of values, but if either is a prime number, you can make the process a lot easier. A good strategy is to divide the number by 2, then by 3, then by 5, gradually moving to larger primes until you find a perfect divisor.
- Example: If you don't know any factor of 24, try dividing it by a small prime number. You start with 2 and you get 24 = 2 x 12. You haven't finished the job yet, but it's a good place to start.
- Since 2 is a prime number, it's a good divisor to start with when you're breaking down an even number.
Step 5. Set up a breakdown scheme
This is a graphical method that helps you organize the problem and track factors. To begin, draw two "branches" that divide from the original number, then write down the first two factors at the other end of those segments.
- Example:
- 24
- /\
- 2 12
Step 6. Proceed with further breaking down the numbers
Look at the pair of values you found (the second row of the pattern) and ask yourself if both are prime numbers. If one of them is not, you can divide it further by always applying the same technique. Draw two more branches starting from the number and write another pair of factors in the third row.
- Example: 12 is not a prime number, so you can factor it further. Use the value pair 12 = 2 x 6 and add it to the pattern.
- 24
- /\
- 2 12
- /\
- 2 x 6
Step 7. Return the prime number
If one of the two factors in the previous line is a prime number, rewrite it in the one below using a single "branch". There is no way to break it down further, so you just need to keep track of it.
- Example: 2 is a prime number, bring it back from the second to the third line.
- 24
- /\
- 2 12
- / /\
- 2 2 6
Step 8. Proceed like this until you get only prime numbers
Check each line as you write it; if it contains values that can be split, proceed by adding another layer. You have finished the decomposition when you find yourself only with prime numbers.
- Example: 6 is not a prime number and must be divided again; 2 instead is, you just need to rewrite it in the next line.
- 24
- /\
- 2 12
- / /\
- 2 2 6
- / / /\
- 2 2 2 3
Step 9. Write the final line as a sequence of prime factors
Eventually, you will have numbers that can be divided by 1 and by themselves. When this happens, the process is finished and the sequence of prime values that makes up the starting number must be rewritten in the form of multiplication.
- Check the work done by multiplying the numbers that make up the last row; the product should match the original number.
- Example: the final line of the factoring scheme contains only 2s and 3s; both are prime numbers, so you have finished the decomposition. You can rewrite the starting number in the form of multiplying factors: 24 = 2 x 2 x 2 x 3.
- The order of the factors is not important, even "2 x 3 x 2 x 2" is correct.
Step 10. Simplify the sequence using powers (optional)
If you know how to use exponents, you can express the prime factorization in a way that is easier to read. Remember that a power is a number with a base followed by a exponent which indicates the number of times you have to multiply the base by itself.
Example: In the 2 x 2 x 2 x 3 sequence, determine how many times the number 2 appears. Since it repeats 3 times, you can rewrite 2 x 2 x 2 as 23. The simplified expression becomes: 23 x 3.
Part 2 of 2: Exploiting Prime Factor Breakdown
Step 1. Find the greatest common divisor of two numbers
This value (GCD) corresponds to the largest number that can divide both numbers under consideration. Below, we explain how to find the GCD between 30 and 36 using the prime factorization:
- Find the prime factorization of the two numbers. The decomposition of 30 is 2 x 3 x 5. That of 36 is 2 x 2 x 3 x 3.
-
Find the number that appears in both sequences. Delete it and rewrite each multiplication in a single line. For example, the number 2 appears in both decompositions, you can delete it and return only one to the new line
Step 2.. Then we have 30 = 2 x 3 x 5 and 36 = 2 x 2 x 3 x 3.
-
Repeat the process until there are no more common factors. In the sequences there is also the number 3, then rewrite it on the new line to cancel
Step 2
Step 3.. Compare 30 = 2 x 3 x 5 and 36 = 2 x 2 x 3 x 3. There are no other common factors.
-
To find the GCD multiply all the shared factors. In this example there is only 2 and 3, so the greatest common factor is 2 x 3 =
Step 6.. This is the largest number which is a factor of both 30 and 36.
Step 2. Simplify the fractions using the GCD
You can exploit it whenever a fraction is not reduced to a minimum. Find the greatest common factor between the numerator and denominator as described above and then divide both sides of the fraction by this number. The solution is a fraction of equal value, but expressed in the simplified form.
- For example, simplify the fraction 30/36. You have already found the GCD which is 6, so proceed with the divisions:
- 30 ÷ 6 = 5
- 36 ÷ 6 = 6
- 30/36 = 5/6
Step 3. Find the least common multiple of two numbers
This is the minimum value (mcm) which includes both numbers in question among its factors. For example, the lcm of 2 and 3 is 6 because the latter has both 2 and 3 as factors. Here's how to find it with factoring:
- Begin factoring the two numbers into prime factors. For example, the sequence of 126 is 2 x 3 x 3 x 7, while that of 84 is 2 x 2 x 3 x 7.
- Check how many times each factor appears; choose the sequence in which it is present several times and circle it. For example, the number 2 appears once in the decomposition of 126, but twice in that of 84. Circle 2 x 2 in the second list.
-
Repeat the process for each individual factor. For example, the number 3 appears in the first sequence more frequently, so circle it 3 x 3. The 7 is only present once in each list, so you only have to highlight one
Step 7. (in this case it doesn't matter which sequence you choose it from).
- Multiply all the circled numbers together and find the least common multiple. Considering the previous example, the lcm of 126 and 84 is 2 x 2 x 3 x 3 x 7 = 252. This is the smallest number that has both 126 and 84 as factors.
Step 4. Use the least common multiple to add fractions
Before proceeding with this operation, you must manipulate the fractions so that they have the same denominator. Find the lcm between the denominators and multiply each fraction so that each has just the least common multiplier as the denominator; once you have expressed the fractional numbers in this way, you can add them together.
- For example, suppose you need to solve 1/6 + 4/21.
- Using the method described above, you can find the lcm between 6 and 21 which is 42.
- Transform 1/6 into a fraction with a denominator of 42. To do this, solve 42 ÷ 6 = 7. Multiply 1/6 x 7/7 = 7/42.
- To transform 4/21 In a fraction with a denominator of 42, solve 42 ÷ 21 = 2. Multiply 4/21 x 2/2 = 8/42.
- Now the fractions have the same denominator and you can easily add them: 7/42 + 8/42 = 15/42.
Practical Problems
- Try to solve the problems proposed here by yourself; when you believe you have found the correct result, highlight the solution to make it visible. The latter problems are more complex.
- Prime 16 into prime factors: 2 x 2 x 2 x 2
- Rewrite the solution using the powers: 24
- Find the factorization of 45: 3 x 3 x 5
- Rewrite the solution in the form of powers: 32 x 5
- Factor 34 into prime factors: 2 x 17
- Find the decomposition of 154: 2 x 7 x 11
- Factor 8 and 40 into prime factors and then calculate the greatest common factor (divisor): The decomposition of 8 is 2 x 2 x 2 x 2; that of 40 is 2 x 2 x 2 x 5; the GCD is 2 x 2 x 2 = 6.
- Find the prime factorization of 18 and 52, then calculate the least common multiple: The decomposition of 18 is 2 x 3 x 3; that of 52 is 2 x 2 x 13; the mcm is 2 x 2 x 3 x 3 x 13 = 468.
Advice
- Each number can be factored into a single sequence of prime factors. No matter what intermediate factors you use, you will eventually get that specific representation; this concept is called the fundamental theorem of arithmetic.
- Instead of rewriting the primes at each step of the decomposition, you can just circle them. When finished, all numbers marked with a circle are prime factors.
- Always check the work done, you could make trivial mistakes and not notice it.
- Watch out for "trick questions"; if you are asked to factor a prime number into prime factors, you do not need to do any calculations. The prime factors of 17 are simply 1 and 17, you don't have to do any further subdivision.
- You can find the greatest common factor and the least common multiple of three or more numbers.
