| Math Mojo - Making Math Meaningful |
This was the question:
What's the prime factorization of 1 trillion using exponents?
(This lesson will be more than just about that. It will also be about basic prime factorization. The question automatically leads you in to it. Stick with the whole lesson and learn some cool stuff, ok?)
Professor Homunculus' answer:
Here is the simple answer (it will be followed by the very long explanation):
The prime factors of ANY power of 10 are the same as (2x5) to the same power.
Try it and see.
So any time you see a number like 10, 100, 1,000 etc, you can immediately
realize that it is (2 x 5) to whichever power of ten the original number was.
And by the distributive power, (2x5) to whatever power is the same as 2 to that
power x 5 to that power.
For example: 100 = (2x5)^2 so the prime factors of 100 are 2^2 x 5^2.
(For purposes of displaying
exponents on the web, the "^" sign is used to mean "to the power
of…" (Normally we write the exponent small, and up to the right of
the base, but it is a pain to get numbers to display like that on the web, so
mathematicians and web-designers have agreed that you can just as well use the
"^" sign.)
Basically, you count the zeros, and use that number as the exponent of 2 x 5.
So 100,000 would be 2^5 x 5^5, and 1,000,000 would be 2^6 x 5^6.
That was the short answer.
Here is the long explanation:
Now let's get to the answer to your question:
You start with the lowest prime number,
and divide it into a trillion.
Keep dividing by two as often as you can, and record the amount of times it
goes in evenly. That will be the exponent of two in the final factorization.
and so on.
(DON"T USE A CALCULATOR* to do simple things like divide a number by 2! Take the time now to train yourself to be able to do it in your mind and just write the answer. If you have to in the beginning, do it on paper by short division. For pity's sake don't write out those stupid long divisions anymore! (Teachers who didn't teach you short division are going to the same hot place after they die that those pedophiliac priests are going.)
If you keep going with those divisions by 2, you will find that after you have
divided 1 trillion by 2^12, you will have gotten 244,140,625. That is not divisible
by 2 anymore, so you have to try the next highest prime number.
Once you have exhausted a prime number, that number will never divide into the
remainder again, so you can forget about having to divide by 2 any more for
this example. So you write the beginning of your answers as 2^12.
The next highest prime number is 3. If you know your divisibility rules, you
can tell at a glance that the above number is not divisible by 3.
4 is not prime.
So divide by 5 and get 48,828,125.
Just as with the 2s, keep track of the amount of times you divided by 5 (or
any other primes in the future), and write that as the exponent of the number.
So far we have 5^1. So, so far the answer is 2^12 x 5^1, but we keep
going until the final division gives us the number 1. (We still have a long
way to go, but always with the same system, so all you need is patience; you
already have all the tools you need.)
48,828,125 / 5 = 9,765,625 (that makes 2^12 x 5^2)
9,765,625 / 5 = 1,953,125 (that makes 2^12 x 5^3)
1,953,125 / 5 = 390,625 (that makes 2^12 x 5^4)
390,625 / 5 = 78,125 (that makes 2^12 x 5^5)
78,125 / 5 = 15,625 (that makes 2^12 x 5^6)
15,625 / 5 = 3,125 (that makes 2^12 x 5^7)
3,125 / 5 = 625 (that makes 2^12 x 5^8)
125 / 5 = 125 (that makes 2^12 x 5^9)
125 / 5 = 25 (that makes 2^12 x 5^10)
25 / 5 = 5 (that makes 2^12 x 5^11)
5 / 5 = 1 (that makes 2^12 x 5^12)
Now you have finally reached the number 1, so you are done.
1 trillion factored into it's prime factors is 2^12 x 5^12.
One more thing which would have made the problem much easier for you is if you had memorized the powers of 2 to at least the tenth power. Sounds far-fetched, but it is very easy to do, and you will have use of it many, many times even if you merely finish high school math. It is great to know for standardized tests, too. Also knowing the powers of five up to the fourth power will come in handy.
If you think that is too much, you should think again. Just learning some simple
things like that will set you apart from the kids who merely "pass"
math. It will help get you "into" more interesting things about math
when you see what can be made easier with such simple knowledge.
Can you imagine not knowing your times-tables up to 10? You know how far ahead
of kids who don't know them that you are already? That's about how far beyond
you will be of kids who don't know their exponent "tables" once you
learn them, and a few things you can do with them.
One more thought about this: If you know your exponents of 5 up to 3, (5^3 is
125) you can use simple speedmath methods to divide by 125 (in your head!) and
that will save you a lot of work.
You can find more about prime factors in the "Pretty Good Guide to Prime Factorization."
* Calculators were invented by vampires to suck your brains out. Don't use them for simple math!
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