初等代数学:
素因数分解

1.  
136136
  解答
2.  
4848
  解答
3.  
7272
  解答
4.  
5454
  解答
5.  
140140
  解答
6.  
448448
  解答
7.  
504504
  解答
8.  
256256
  解答
9.  
130130
  解答
10.  
324324
  解答

Prime Factorization - Introduction

All whole numbers can be reduced to factors — a set of smaller numbers that when multiplied together yield the original answer. For example,
100100
can be broken down into several sets of factors including:
100=10×10100=10\times 10

100=20×5100=20\times 5

100=5×5×2×2100=5\times 5\times 2\times 2
Certain numbers, called prime numbers cannot be evenly divided by any number except
11
. An example of a prime number is
1717
, which has only two factors:
11
and
1717
.
Combining the two concepts yields prime factorization, which is the process of breaking down a large number into the set of prime numbers that multiply together to create the original number.

How Does Prime Factorization Work?

Prime factorization is not that complicated once you know the steps:

1. Start with a number. For this example, let this be
777777
.

2. Divide
777777
by the smallest prime number that yields a whole number.

First, try
22
:
777÷2=338.5777\div 2=338.5
This doesn't work because the result has the decimal. It is not a whole number.
Next, try
33
:
777÷3=259777\div 3=259
Therefore,
33
is our first prime factor.

3. Divide
259259
by the smallest prime number that yields a whole number.

Start again with
22
.
259÷2=129.5259\div 2=129.5
259÷3=86.333333259\div 3=86.333333
259÷5=51.8259\div 5=51.8
259÷7=37259\div 7=37
Therefore,
77
is our second prime factor.

4. Since 37 is a prime number, we are done.

Combining the answers from our previous steps gives the result: The prime factorization of
777777
is:
777=3×7×37777=3\times 7\times 37

The General Rule

Ready to try some prime factorization practice? To sum up, the general rule is, start with the smallest prime, which is 2. If the number won’t evenly divide, move up in the list of primes to 3, 5, 7, 11, etc. Stop until you find one that evenly divides. And if you find none, this means you are dealing with a prime number.

What's Next

Prime factorization is an example of how students can break down a complex problem into smaller, more manageable pieces. Please also try other practice problems at the top of this page. They are designed to help challenge students while also providing step-by-step solutions. Dealing with a particularly large number? Use Cymath as your prime factorization calculator to quickly get the answer you need.
It’s all part of the Cymath advantage — one of the most popular math apps combined with concise problems and solutions to help students improve their math skills.
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