How to Parse a Number Into Factories

The factors of a given number are numbers that, when multiplied together, have the same product as the previous given number. Think of it another way, every number is a product of many factors. Learning to factorize – or break a number into factors – is an important math skill that applies not only in basic arithmetic, but also in algebra, calculus and more. See Step 1 to start learning how to factor a number!

Table of Contents

Factoring Basic Integers

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Write your number. To start the analysis, you need a number – any number, but for the sake of the article, let’s start with a simple integer. Integers are numbers that do not have fractions or decimals (integers consisting of all positive and negative integers).

Please choose number 12 . Write this number down on scratch paper.

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Find two more numbers whose product is the original number you chose. Any integer can be written as the product of two other integers. Even prime numbers can be written as the product of 1 and itself. Thinking of a number as the product of two factors can cause you to think “backwards” – you must have wondered, “what multiplication results in this number?”

For our example, 12 has several factors like 12 × 1, 6 × 2, and 3 × 4 all equal 12. So we can say that the factor of 12 is 1, 2, 3, 4 , 6, and 12 . Let’s use the factors of 6 and 2 for the purpose of the lesson.
Even numbers are especially easy to parse because every even number has a factor of 2. 4 = 2 × 2, 26 = 13 × 2, etc.

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Determines whether the current factors can be further analyzed. A lot of numbers – especially large ones – can be parsed a few more times. Once you have found the two factors of a given number, if a factor itself has its own factor, you can also factor this into smaller factors. Depending on the case, the analysis may or may not be beneficial.

According to our example, the number 12 has been decomposed into 2 × 6. Notice that 6 also has its own factor – 3 × 2 = 6. So we can say that 12 = 2 × (3 × 2) .

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Stop the analysis when all the factors are prime. Prime numbers are numbers that are only divisible by 1 and itself. For example, 2, 3, 5, 7, 11, 13 and 17 are prime numbers. Once you have analyzed some product of prime factors, further analysis is redundant. Further factoring these is its own product and one has no effect, so you can stop.

In our example, 12 has been parsed as 2 × (2 × 3). 2, 2, and 3 are all prime numbers. If further analysis, we have to parse as (2 × 1) × ((2 × 1)(3 × 1)), which usually has no effect and is ignored.

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Analyze negative numbers in the same way. The way to analyze negative numbers is almost similar to the way to analyze positive numbers. The only difference is that the product of the factors must be a negative number, so the number of negative factors must be odd.

For example, let’s analyze -60. Accordingly:
- -60 = -10 × 6
- -60 = (-5 × 2) × 6
- -60 = (-5 × 2) × (3 × 2)
- -60 = -5 × 2 × 3 × 2 . Note that as long as the number of negative factors is odd, the product of all the factors will also be negative, similar to when there is only one negative factor. For example, -5 × 2 × -3 × -2 is also -60.

How to Factor Large Numbers

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Write your number above a table with 2 columns. Factoring small numbers is usually quite simple, but factoring large numbers is more complicated. Most of us will have a hard time factoring a 4 or 5 digit number into prime numbers without using pen and paper. Fortunately, when you draw the board, the process becomes a lot easier. Write your numbers above the T with two columns – you will use this table to keep track of the growing list of factors.

For our example, let’s choose a 4-digit number to factor, which is 6.552 .

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Divide your number by the smallest possible prime factor. Divide your number by the smallest prime factor (other than 1) that your number is divisible by and leave no remainder. Write the prime factor in the left column and write the quotient of the horizontal division in the right column. As noted above, even numbers are easier to parse because their smallest prime factor is always 2. On the other hand, odd numbers will have the smallest prime factor other than 2.

In our example, since 6.552 is an even number, we know that 2 is the smallest prime factor of this number. 6.552 ÷ 2 = 3.276. In the left column, we write 2 , and 3.276 in the right column.

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Continue factoring in this way. Next, divide the number in the right column by its smallest prime factor, instead of using the number above the table. Write the selected prime factor in the left column and the new result of the division in the right column. Continue this process – after each repetition, the numbers in the right column will get smaller and smaller.

Let’s continue the analysis. 3.276 ÷ 2 = 1.638, so we’ll write an extra 2 at the bottom of the left column, and 1.638 at the bottom of the right column. 1.638 ÷ 2 = 819, so we will write 2 and 819 at the bottom of the two columns as before.

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Analyze the odd number by trying to divide it by small prime factors. Finding the smallest prime factor of odd numbers is harder than even numbers because they don’t automatically have 2 as the smallest prime factor. When you have an odd number, try dividing it by some other small primes 2 – 3, 5, 7, 11, and so on until the odd number is divisible by a prime and zero. leave balance. That is the smallest prime factor.

For our example, we get 819. 819 is odd, so 2 is not a factor of 819. Instead of writing 2, we’ll try the next prime number: 3. 819 ÷ 3 = 273 and there is no remainder, so we write 3 and 273 .
When guessing factors, you should try all primes whose value is less than or equal to the square root of the largest factor you have found. If your number isn’t divisible by any of the factors, you’re probably trying to factor a prime, and the factoring process could end there.

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Continue until the quotient is 1. Continue dividing the number in the right column by its smallest prime factor until you have a prime in the right column. Divide this number by itself – this step will record the number in the left column and “1” in the right column.

Let’s finish our numerical analysis. See detailed explanation below:
- Divide by 3: 273 ÷ 3 = 91, no remainder, so we write 3 and 91 .
- Let’s go on to the number 3: 3 is not a factor of 91, and the next smallest prime number (5) is also not a factor of 91, however 91 ÷ 7 = 13, there is no remainder, so we write 7 and 13 .
- Let’s keep trying with 7: 7 is not a factor of 13, so is 11 (the next prime number), but 13 has its own factor: 13 ÷ 13 = 1. So to complete the table analysis, we write 13 and 1 . We can stop the analysis here.

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The numbers in the left column are the factors of the number you originally selected. When the right column ends with a 1, you’re done. The numbers in the left column are the factors you need to find. In other words, the product of those numbers will be equal to the number listed above the board. If these factors are repeated many times, you can use exponentiation notation to save space. For example, if your factoring series has four 2s, you can write 2 ⁴ instead of 2 × 2 × 2 × 2.

In our example, 6.552 = 2 ³ × 3 ² × 7 × 13 . This is the complete result after factoring 6,552 into primes. Regardless of the order in which the multiplication is performed, the final product will be 6.552.

Advice

An important point is the concept of prime numbers: a number that has only two factors, 1 and itself. 3 is prime because its factors are only 1 and 3. Otherwise, 4 has another factor of 2. A number that is not prime is called a composite number . (The number 1 itself is neither prime nor composite — that’s a special case.)
The smallest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and 23.
Understand that a number is considered a factor of a larger number if the larger number is “divisible by the smaller” – that is, the larger number is divisible by the smaller number and leaves no remainder. For example, 6 is a factor of 24, because 24 ÷ 6 = 4 and there is no remainder. Conversely, 6 is not a factor of 25.
Some numbers can be parsed in a faster way, but the above approach always works, plus, the prime factors are listed in ascending order as you’re done.
Remember we are only talking about “natural numbers” – sometimes called “counts”: 1, 2, 3, 4, 5… We won’t go into negative numbers or fractions, That can be covered in separate articles.
If the sum of the digits of a number is divisible by three, then three is a factor of that divisor. (819 has the sum of the digits 8+1+9=18, 1+8=9. Three is a factor of nine so it is also a factor of 819.)

Warning

Don’t do unnecessary things. Once you’ve eliminated a factor value, you don’t need to try again. Once we are sure that 2 is not a factor of 819, we don’t need to retry with 2 for the rest of the process.

Things you need

Paper
Pens write, it is recommended to use pencils and erasers
Computer (optional)

Steps

Factoring Basic Integers

How to Factor Large Numbers

Advice

Warning

Things you need

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