Unfortunately there's no magic formula that will get you the prime factors each time, but you can work through the process systematically! :)

To find the prime factors of a large number, you can make something called a "factor tree"—perhaps you learned about this when you were younger, or perhaps you've come across it as you try to master prime factorization for a test. Let's find the prime factors of 14000 to show how this works.

**Step 1: Find any two numbers, any at all, that multiply to make 14000. **

Make it easy! If you can only think of a factor like 2 or 3, that is completely fine! I'm just going to pick

14000 = 14 * 1000 for my first set of factors.

**Step 2: With each factor we just found, repeat that process.**

Express each factor from the first step as as the product of two smaller factors. I had 14000 = 14 * 1000, so now I want to break apart both 14 and 1000.

- 14 = 2 * 7
- 1000 = 10 * 100

So, now I can combine these newly broken apart factors. My tree is now coming along nicely:

14000 = 14 * 1000

14000 = (2 * 7) * (10 * 100)

**Step 3: Continue to repeat Step 2 until you have a set of only prime factors.**

- 2 = prime
- 7 = prime
- 10 = 2 * 5
- 100 = 10 * 10

14000 = 14 * 1000

14000 = (2 * 7) * (10 * 100)

14000 = (2 * 7) * (2 * 5) * (10 * 10)

14000 = (2 * 7) * (2 * 5) * (2 * 5) * (2 * 5)

**Step 4: Group all of the prime factors together and rewrite with exponents if desired.**

14000 = 2 * 2 * 2 * 2 * 5 * 5 * 5 * 7

14000 = (2^4) * (5^3) * (7)

To find the exponents, you see how many of each number there are (for example, 2 * 2 * 2 * 2 = 2^4 because there are four 2s). If you do this systematically, you can easily find all the prime factors of large numbers and then use the prime factors to solve a variety of problems about factors. :)

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