2013 AMC 10 A #21 / AMC 12 A #17

- All right, what we've got here are 12 pirates. They're gonna divide up a treasure chest of gold. And here's how they're gonna do it. First pirate's gonna come along and take 1/12 of the gold that's in the chest. Second pirate's gonna come along, take 2/12 of the whatever's left after the first pirate is finished. Third pirate's gonna take 3/12 of whatever's left after the second pirate finished, and on, and on, and on. And 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, positive whole number of coins using this process. And we're gonna start with x 'cause x marks the spot. x is the number of coins that was in the chest at the beginning. And the first pirate comes along and takes 1/12, now there's 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, there's this much. She's gonna leave 10/12 of this amount for the next pirate. 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 wanna figure out, how much does this, the last pirate, receive? So we wanna figure out what the value of this expression is. And we can write this a lot shorter as x times 11 factorial over 12 to the 11th. And we wanna figure out what this is. Now, x is the smallest value that makes this an integer. Well, actually x is the smallest value that makes sure each pirate gets an integer number of coins. Don't worry about that right now. I'm just gonna worry about the last pirate and figure out what the last pirate gets. So, well if x, if you just choose x is 12 to the 11th, well then this will 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 two and three from this 11 factorial, see what's left. We're gonna be left with a factor of 11. And then we're gonna have two fives from the five and the 10. And we're gonna have a seven sitting in there. And then we need to figure out, well, we're gonna simplify this fraction, take out factors of two. I mean 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 twos in 11 factorial, the number of factors of two, you get two, four, six, eight, 10, that's five. You get an extra one from the four, two extra ones from the eight. Eight factors of two up here. There are 22 down there. That leaves us two to the 14th. And the factors of three. You have three, six, and nine up there. You get an extra factor of three in the nine. Four factors of three up there. Eleven down there. That leaves us three to the seventh. Now we can go ahead and multiply out this numerator seven times 11. That's 77 times 25. Well, 80 times 25, that's 2,000. So 77 times 25 is going to be 1925, and that makes us happy, 'cause 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, 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 two somewhere. So one way to check your answer is just to work it through. Let x be two to the 14th times three to the seventh and see if each pirate gets a positive whole number of coins. Now if you're jammed on time on a test, you're gonna bubble D and move on, and that's probably the right thing to do, 'cause it sure looks like you're gonna get a positive, you're gonna get a whole number of coins at each step. But let's just check it real quick. We start off with two to the 14th times three to the seventh coins. What happens? We wanna make sure we end up with 1925 at the end, and we wanna make sure each pirate has a positive whole number of coins, so here we go. We start off with two to the 14th times three to the seventh coins. I'm gonna box this off 'cause we're gonna a need a little space here. First pirate's gonna take 1/12 of that. That's gonna work out just fine, and leave 11 times two, 11/12 of it, 11 times two to the 12th times three to the sixth. Now you see why we have to start worrying. We're gonna be chippin' away at these powers of two and three. We just don't ever wanna end up with a fractional number of coins. Next pirate comes along, he needs to take 2/12, 1/6 of this. That's gonna work out just fine, and gonna leave 5/6 of this amount, leaving 5/6 of this gives us a factor of five there, but it's gonna take away another factor of two, another factor of three. Next person comes along, takes 1/4 of the coins, leaves 3/4 of the coins. So that's gonna throw another factor of three back in, but take two factors of two away. So this is how may 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 three, throws in another factor of two. Next one's gonna come along, take 5/12, leave 7/12, so that's seven times 55. Now we're knockin' out two factors of two and a factor of three. Next one's gonna come along, take half, leave half. That one's pretty easy, just knocks out another factor of two. Next one comes along, takes 7/12, leaves 5/12, and we know what five times seven times 55 is now. We've already computed that. 1925 times two to the fifth, times three cubed. Next one comes in and takes away 8/12, leaving 4/12 of this amount, so leaving 1/3 of this, leaving 1925 times two to the fifth times three 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, is taking 10/12, leaving 2/12, leaving 1/6 of this amount. As you can see, at each step, the pirate who's walkin' away with the loot is walking away with an integer number of coins. And finally here at the end, the last one's gonna come in and take 11/12 of this which is gonna work out just fine. You're gonna get a whole number of coins because we have that factor of 12 right there, and we're gonna 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.