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Can you help me find the prime factors of large numbers?

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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