Another Coconut/Monkey example

Another Coconut/Monkey example

agrotz2 15 Feb 2009 22:45

I came across this problem and found it to be similar to the one we had with the monkeys and the coconuts that we all loved so much! So I figured I'd post it to see if someone wanted to try to work it out.

An old woman goes to market and a horse steps on her basket and crashes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, there was one egg left. The same happened when she picked them out three, four, five, and six at a time, but when she took them seven at a time they came out even. What is the smallest number of eggs she could have had?

The answer is 301, and the solution can be found on the website provided. I just thought it was another "story" problem that could be a little practice for us.

http://www.cut-the-knot.org/blue/chinese.shtml

Re: Another Coconut/Monkey example

Aviva Calamaro 16 Feb 2009 02:28

I tried doing the problem, and then I looked at the answer on the website. I understand the second half of their work, but how do they get 1 mod 60 from the first five statements? We talked in class about how to solve a problem where the gcd is not one, but that was only with two congruence equations. How do you solve it with multiple congruence equations?

Re: Another Coconut/Monkey example

Ryan Schipfer 16 Feb 2009 04:34

if you apply the CRT to q2,q3,q4 (most of it is actually in the example on pg 56 in the textbook except our $b_i$s our equal to one), it works out to $x \equiv 1 \mod 60$. Not sure why they use all at once.

However, I found another example of a CRT problem similar to the coconut one. This one is about pirates!

An ancient Chinese problem asks for the least number of gold coins a band of 17 pirates could have stolen. The problem states that when the pirates divided the coins into equal piles, 3 coins were left over. When they fought over who should get the extra coins, one of the pirates was slain. When the remaining pirates divided the coins into equal piles, 10 coins were left over. When the pirates fought again over who should get the extra coins, another pirate was slain. When they divided the coins in equal piles again, no coins were left over. What is the answer to this problem?

Re: Another Coconut/Monkey example

Aviva Calamaro 16 Feb 2009 17:32

I got that the answer is 3930. (That's a lot of coins!)

x congruent to 3 (17).
x cong to 10 (16).
x cong to 0 (15).

Using CRT, N1 = 16x15, X1 = 9 (17), a1 = 3
N2 = 17x15, X2 = -1 (16), a2 = 10
N3 = 17x16, X3 = 8 (15), a3 = 0 (so this part just goes away).