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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.