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2013 AMC 10 A #21 / AMC 12 A #17

Video transcript
All right. What we've got here are 12 pirates. They're going to divide out a treasure chest of gold. And here's how they're going to do it. First pirate's going to come along, take 1/12 of the gold that's in the chest. Second pirate's going to come along, take 2/12 of whatever's left after the first pirate is finished. Third pirate's going to take 3/12 of whatever's left after the second pirate finished, and on, and on, and on. Let's see what happens here. Each pirate gets a positive whole number of coins. And the number of coins that was in the chest is the smallest number of coins for which it's possible for each pirate to get, a positive number of coins, a positive whole number of coins using this process. We're going to start with x because x marks the spot. x is the number of coins that was in the chest at the beginning. And the first pirate comes along, takes 1/12. That leaves 11/12 remaining for the next pirate who comes along. And the next pirate, second pirate takes 2/12 and leaves 10/12 of what was there when she got there for the next pirate. So she gets there and there's this much. She's going to leave 10/12 of this amount for the next pirate. The next pirate comes along, takes 3/12, leaves 9/12 of this for the following pirate, and on, and on, and on we go until we get to the last few pirates. The 11th pirate takes 11/12, leaves 1/12 of what was there for the last pirate, who comes along and takes everything that's left. Well, that's what we want to figure out. How much does the last pirate receive? So we want to figure out what the value of this expression is. We can write this a lot shorter as x times 11 factorial over 12 to the 11th. And we want to figure out what this is. Now, x is the smallest value that makes this an integer. Actually, x is the smallest value that makes sure each pirate gets an integer number of coins. I'm not going to worry about that right now. I'm just going to worry about the last pirate and figure what the last pirate gets. If you just choose x is 12 to the 11th, well, this will just come out to be an integer. But 11 factorial is not any of these choices. So we can simplify this fraction. We can take out all the factors of 2 and 3 from this 11 factorial and see what's left. We're going to be left with a factor of 11. And then we're going to have two 5's from the 5 and the 10. And we're going to have a 7 sitting in there. And then we need to figure out well, we're going to simplify this fraction, take out factors of 2. We could stop right here, just compute this and call that the answer. But I'm a little bothered by that whole every pirate has to get an integer number of coins thing. But let's go ahead and simplify this fraction. The number of 2's in 11 factorial, the number of factors of 2, you get 2, 4, 6, 8, 10. That's 5. You get an extra one from the 4, two extra ones from the 8. 8 factors of 2 up here are 22 down there. That leaves us 2 to the 14th. And then the factors of 3, you have 3, 6, and 9 up there. You get an extra factor of 3 in the 9. Four factors of 3 up there. 11 down there. That leaves 3 to the seventh. Now we can go ahead and multiply out this numerator. 7 times 11 is 77 times 25. Well, 80 times 25, that's 2000. So 77 times 25 is going to be 1925. And that makes us happy because 1925 is sitting right there. But you'd be forgiven for just circling the 1925, calling it D, and moving on. But seeing that 3850 right there, that scares me a little bit. It makes me remember that each pirate has to get a positive whole number of coins, so maybe there's a reason we have to multiply by 2 somewhere. So one way to check your answer is just to work it through. Let x be 2 to the 14th times 3 to the seventh. And then see if each pirate gets a positive whole number of coins. Now if you're jammed on time on a test, you're going to bubble D and move on. And that's probably the right thing to do because it sure looks like you're going to get a whole number of coins at each step. But let's just check it real quick. If we start off with 2 to the 14th times 3 the seventh coins, what happens? We want to make sure we end up with 1925 at the end. And we want to make sure each pirate has a positive whole number of coins. So here we go. We start off with 2 to the 14th times 3 to the seventh coins. I'm going to box this off because we're going to need a little space here. First pirate's going to take 1/12 of that. That's going to work out just fine and leave 11 times 2-- 11/12 of it, 11 times 2 to the 12 times 3 to the 6. And now you see why we have to start worrying. We're going to be chipping away at these powers of 2 and 3. We just don't ever want to end up with a fractional number of coins. Next pirate comes along. He gets to take 2/12, 1/6 of this. That's going to work out just fine. And gonna leave 5/6 of this amount. Leaving 5/6 of this gives us factor of 5 there, but it's going to take away another factor of 2, another factor of 3. Next person comes along, takes 1/4 of the coins, leaves 3/4 of the coins. So that's going to throw another factor of 3 back in, but take two factors of 2 away. So this is how many coins we have left. Then the next one comes in and takes 1/3. That's 4/12. Leaves 2/3 of this amount, takes away a factor of 3, throws in another factor of 2. Next one's going to come along, take 5/12, leave 7/12. So that's 7 times 55. Now we're knocking out two factors of 2 and a factor of 3. Next one's going to come along, take 1/2, leave 1/2. That one's pretty easy, just knocks out another factor of 2. Next one comes along, takes 7/12, leaves 5/12. And we know what 5 times 7 times 55 is now. We've already computed that, 1925, times 2 the fifth times 3 cubed. Next one comes in and takes away 8/12, leaving 4/12 of this mess. So leaving 1/3, leaving 1925 times 2 to the fifth times 3 squared. Whew. Almost there. Next one comes along, takes 3/4, leaves 1/4 of this amount. And then the next one comes along. That's this one right here. It's taking 10/12, leaving 2/12, leaving 1/6 of this amount. As you can see at each step, the pirate who's walking away with the loot is walking away with an integer number of coins. And finally here at the end, the last one's going to come along and take 11/12 of this, which is going to work out just fine. You're going to get a whole number of coins because we have that factor of 12 right there. And we're going to leave 1/12 remaining which is the 1925 coins. So if you had just bubbled D and moved on, it all worked out just fine. And we're done.