The factors of a number are the digits which, when multiplied together, give the number itself as a product. To better understand the concept, you can consider each number as the result of multiplying its factors. Learning to factor a number into prime factors is an important mathematical skill that will be useful not only for arithmetic problems, but also for algebra, mathematical analysis and so on. Read on to learn more.
Steps
Method 1 of 2: Factoring the Basic Integers
Step 1. Write down the number under consideration
To start the decomposition you can use any number but, for our educational purposes, we use a simple integer. An integer is a number with no decimal or fractional component (all integers can be negative or positive).
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We choose the number
Step 12.. Write it on a piece of paper.
Step 2. Find two numbers which, when multiplied together, give the original number
Each integer can be rewritten as the product of two other integers. Even the prime numbers can be considered the product of themselves and 1. Finding the factors requires a "backward" reasoning, in practice you have to ask yourself: "which multiplication results in the number under consideration?".
- In the example we have considered, 12 has many factors. 12x1; 6x2; 3x4 all result in 12. So we can say that the factors of 12 are 1, 2, 3, 4, 6 and 12. Again for our purposes, we use factors 6 and 2.
- Even numbers are particularly easy to break down because 2 is a factor. In fact 4 = 2x2; 26 = 2x13 and so on.
Step 3. Check if the factors you have identified can be broken down further
Many numbers, especially large ones, can be broken down many times. When you find two factors of a number that are in turn the product of other smaller factors, you can break it down. Depending on the type of problem you need to solve, this step may or may not be helpful.
In our example, we have reduced 12 to 2x6. 6 also has its own factors (3x2). Then you can rewrite the decomposition as 12 = 2x (3x2).
Step 4. Stop decomposition when you reach prime numbers
These are numbers divisible only by 1 and by themselves. For example 1, 2, 3, 5, 7, 11, 13 and 17 are all prime numbers. When you've factored a number into prime factors, you can't go any further.
In the example of number 12, we have reached the decomposition of 2x (3x2). The numbers 2 and 3 are all prime, if you wanted to proceed to a further decomposition, you should write (2x1) x [(3x1) x (2x1)] which is not useful and should be avoided
Step 5. Negative numbers break down with the same criteria
The only difference is that the factors must be multiplied in such a way as to obtain a negative number; this means that an odd number of factors must be negative.
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Factor -60 into prime factors:
- -60 = -10x6
- -60 = (-5 x 2) x 6
- -60 = (-5 x 2) x (3 x 2)
- -60 = - 5 x 2 x 3 x 2. Note that the presence of an odd amount of negative digits leads to a negative product. If I had written: 5 x 2 x -3 x -2 you would have got 60.
Method 2 of 2: Steps to Break Down the Big Numbers
Factor a Number Step 6 Step 1. Write the number above a two-column table
Although it is not at all difficult to factor a small number, with very large numbers it is a bit more complex. Most of us would have some difficulty in factoring a 4 or 5 digit number into prime factors. Fortunately, a table makes our work easier. Write the number on top of a “T” shaped table to form two columns. This table helps you to record the list of factors.
For our purposes we choose a 4-digit number: 6552.
Factor a Number Step 7 Step 2. Divide the number by the smallest prime factor
You need to find the smallest factor (other than 1) that divides the number without producing a remainder. Write the first factor in the left column and the quotient of the division in the right column. As we have already said, even numbers are easy to break down because the minimum prime factor is 2. Odd numbers, on the other hand, can have a different minimum factor.
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Returning to the example of 6552, which is even, we know that 2 is the smallest prime factor. 6552 ÷ 2 = 3276. In the left column you will write
Step 2. and in the one on the right 3276.
Factor a Number Step 8 Step 3. Continue following this logic
Now you have to decompose the number in the right column always looking for its minimum prime factor. Write the factor in the left column below the first factor you found and the result of the division in the right column. With each step, the number on the right gets smaller and smaller.
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Let's continue with our calculation. 3276 ÷ 2 = 1638, so in the left column you will write another
Step 2. and in the right column 1638. 1638 ÷ 2 = 819, so write a third
Step 2. And 819, always following the same logic.
Factor a Number Step 9 Step 4. Work with odd numbers to find their smallest prime factors
Odd numbers are more difficult to break down, because they are not automatically divisible by a given prime number. When you get an odd number, you have to try with divisors other than two, such as 3, 5, 7, 11, and so on until you get a quotient with no remainder. At that point you have found the smallest prime factor.
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In our previous example, you have reached the number 819. This is an odd value, so 2 cannot be a factor of it. You have to try the next prime number: 3. 819 ÷ 3 = 273 with no remainder, so write
Step 3. in the left column e 273 in the one on the right.
- When looking for factors, you should try all prime numbers up to the square root of the largest factor found so far. If none of the factors is a divisor of the number, then it is likely that it is a prime number and the decomposition process is considered finished.
Factor a Number Step 10 Step 5. Continue until you get 1 as the quotient
Proceed through the divisions looking for the minimum prime factor each time until you reach a prime number in the right column. Now divide it by itself and write "1" in the right column.
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Complete the breakdown. Read the following for details:
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Divide by 3 again: 273 ÷ 3 = 91 with no remainder, then write
Step 3. And 91.
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Try dividing by 3 again: 91 is not divisible by 3 nor by 5 (the prime factor after 3) but you will find that 91 ÷ 7 = 13 with no remainder, so write
Step 7
Step 13..
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Now try dividing 13 by 7: it is not possible to get a quotient without a remainder. Go to the next prime factor, 11. Again 13 is not divisible by 11. At the end you will find that 13 ÷ 13 = 1. Then complete the table by writing
Step 13
Step 1.. You have completed the breakdown.
Factor a Number Step 11 Step 6. Use the numbers in the left column as factors of the original problem number
When you have reached figure 1 in the right column, you are done. In other words, all the numbers in the left column, if multiplied together, give the starting number as a product. If there are any factors that occur multiple times, then you can use exponential notation to save space. For example, if the list of factors has the number 2 four times, then you can write 24 instead of 2x2x2x2.
The number we have considered can be broken down as follows: 6552 = 23 x 32 x 7 x 13. This is the complete prime factorization of 6552. Regardless of the order you follow to carry out the multiplication, the product will always be 6552.
Advice
- The concept of number is also important first: a number that has only two factors, 1 and itself. 3 is a prime number because its only factors are 1 and 3. 4, on the other hand, has 2 among its factors. A number that is not prime is called composite (the number 1, however, is neither considered prime nor composite: it is a special case).
- The smallest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and 23.
- Remember that a number is factor of another major if it "divides it perfectly" without remainder. For example, 6 is a factor of 24 because 24 ÷ 6 = 4 without any remainder; while 6 is not a factor of 25.
- Remember that we are referring only to the so-called "natural numbers": 1, 2, 3, 4, 5… We will not deal with negative numbers or fractions, for which specific articles are needed.
- Some numbers can be broken down more quickly, but this method always works and, in addition, you will have the prime factors listed in ascending order.
- If the sum of the digits that make up a certain number is a multiple of 3, then 3 is a factor of that number. For example: 819 = 8 + 1 + 9 = 18, 1 + 8 = 9. 3 is a factor of 9, so it's a factor of 819.
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