Hi there! Nice collection of brain exercise!While I'm no smart-ass, I think I found a mistake in 'The Monkey and the Coconut'. The lowest answer is not the LCM minus one, but actually the product of all prime numbers below ten minus one. 1*2*3*5*7-1=209 Try it, it's true.
First you need something divisible by ten. So you start with 2*5.
Now you need something that is also divisible by nine, so you multiply by 3*3 to get 2*5*3*3.
In order to make it divisible by eight (2*2*2), you must multiply by 2*2 and get: 2*3*3*5*2*2.
Then you must make it divisible by seven, so you multiply by that and get: 2*3*3*5*2*2*7.
This is also divisible by six (2*3), five, four (2*2), three and two. So you're done, except you need to subract one. Rearranging the numbers a bit you get:
Theres 10 guys with 2519 coconuts...why does it matter if they all have the same amount? Can't they put it in a pile and eat one at a time...Somehow they all go 9 days without eating even one so they're not all that essential to anything anyway. K im done ._."
I'm stranded on a deserted island and don't know how long I will be here. The blood on the Monkey's coconut represented additional protein. Of course, after guy number one gets conked on the head, there is plenty of fresh meat so you arrive at the question again, why would you want the bloodied coconut.
I think Cristian's explanation is about the simplest, most straightforward. What Cristian is actually doing is finding the lcm of all the numbers from 2 to 10. There are several ways to do this, but here's my favorite.
Write out the prime factorization of each of the numbers:
Now write down each prime factor that appears (2, 3, 5 and 7) and raise each to the highest power that it appears with, and multiply these terms together:
2^3 x 3^2 x 5^1 x 7^1 = 8 x 9 x 5 x 7 = 2,520
P.S. I was going to show an approach to this problem using conguences, but I decided not to because it essentially boils down to the same darn thing. It just looks and sounds a little fancier. =:)
I think Cristian's explanation is about the simplest, most straightforward. What Cristian is actually doing is finding the lcm of all the numbers from 2 to 10. There are several ways to do this, but here's my favorite.
Write out the prime factorization of each of the numbers:
Now write down each prime factor that appears (2, 3, 5 and 7) and raise each to the highest power that it appears with, and multiply these terms together:
2^3 x 3^2 x 5^1 x 7^1 = 8 x 9 x 5 x 7 = 2,520
P.S. I was going to show an approach to this problem using congruences, but I decided not to because it essentially boils down to the same darn thing. It just looks and sounds a little fancier. =:)
[A SECOND ANSWER]
If you consider, what I will call, the non-replaceable bag, the answer is "9."
When the first person wakes up, there are 10 coconuts, and the monkey has one, giving 9 in the pile, 1 less than a number divisible by 10).
When the second person wakes up, there are 9 coconuts (what the monkey did with the first one I don't care. Maybe it broke open when he cracked it over the head of the sailor), and the monkey has 1, leaving 8 in the pile, 1 less than a # divisible by 9.
And so on and so on.
Until the last person wakes up, there are no coconuts in the pile, and the monkey has the last one.
You didn't read the entire question, you left off the fact that the monkey only took one coconut. The first person tries to take it from him, then the monkey kills him. the second person then wakes up and tries to rearrange the piles evenly for 9 people because he sees that the first one to wake up is dead, and notices that the piles are short one, and that the monkey has a bloody coconut. he tries to take it and the monkey kills him as well with the same coconut.
The answer is a lot smaller than anything said thus far..
The answer is 19.
This means they found 20 and were going to divide the pile into ten piles of two.
The first guy gets to the pile and finds 19 which can't be divided evenly by 10.
19 also can't be divided by 9, 8, 7, 6, 5, 4, 3 OR 2.
Any questions? :p
19 may not be divided by 9,8,7,6,5,4,3 or 2 but the part you are missing is that 20 does not divide into 9, 8, 7,6 or 3. You need this because you were always one short of having equal piles
o.k lets think for a minute it says in the question that 10 people collected coconuts in ONE DAY o.k? so the answer says there are 2519 coconuts even if it is true who can collect 252(251.9) coconuts in one day? especially if you have been in a shipwreck i think this one could have been set out better.
Now you need something that is also divisible by nine, so you multiply by 3*3 to get 2*5*3*3.
In order to make it divisible by eight (2*2*2), you must multiply by 2*2 and get: 2*3*3*5*2*2.
Then you must make it divisible by seven, so you multiply by that and get: 2*3*3*5*2*2*7.
This is also divisible by six (2*3), five, four (2*2), three and two. So you're done, except you need to subract one. Rearranging the numbers a bit you get:
2*2*2*3*3*5*7-1 = 8*9*5*7-1 = 2520-1 = 2519
Write out the prime factorization of each of the numbers:
2 = 2^1
3 = 3^1
4 = 2^2
5 = 5^1
6 = 2^1 x 3^1
7 = 7^1
8 = 2^3
9 = 3^2
10 = 2^1 x 5^1
Now write down each prime factor that appears (2, 3, 5 and 7) and raise each to the highest power that it appears with, and multiply these terms together:
2^3 x 3^2 x 5^1 x 7^1 = 8 x 9 x 5 x 7 = 2,520
P.S. I was going to show an approach to this problem using conguences, but I decided not to because it essentially boils down to the same darn thing. It just looks and sounds a little fancier. =:)
Write out the prime factorization of each of the numbers:
2 = 2^1
3 = 3^1
4 = 2^2
5 = 5^1
6 = 2^1 x 3^1
7 = 7^1
8 = 2^3
9 = 3^2
10 = 2^1 x 5^1
Now write down each prime factor that appears (2, 3, 5 and 7) and raise each to the highest power that it appears with, and multiply these terms together:
2^3 x 3^2 x 5^1 x 7^1 = 8 x 9 x 5 x 7 = 2,520
P.S. I was going to show an approach to this problem using congruences, but I decided not to because it essentially boils down to the same darn thing. It just looks and sounds a little fancier. =:)
[A SECOND ANSWER]
If you consider, what I will call, the non-replaceable bag, the answer is "9."
When the first person wakes up, there are 10 coconuts, and the monkey has one, giving 9 in the pile, 1 less than a number divisible by 10).
When the second person wakes up, there are 9 coconuts (what the monkey did with the first one I don't care. Maybe it broke open when he cracked it over the head of the sailor), and the monkey has 1, leaving 8 in the pile, 1 less than a # divisible by 9.
And so on and so on.
Until the last person wakes up, there are no coconuts in the pile, and the monkey has the last one.
This means they found 20 and were going to divide the pile into ten piles of two.
The first guy gets to the pile and finds 19 which can't be divided evenly by 10.
19 also can't be divided by 9, 8, 7, 6, 5, 4, 3 OR 2.
Any questions? :p
"hmmm, a dead guy and a monkey holding a bloody coconut... i think ill try to steal the monkey's coconut... what could go wrong?"