# Understand the problem

##### Source of the problem

##### Topic

Number Theory

##### Difficulty Level

7/10

##### Suggested Book

Elementary Number Theory by David M. Burton

# Start with hints

Well, just give the problem a good read. Probably, with a little bit of thought, you can even get this done without a hint !

We could start this the traditional way, be assuming the number of coins to be **x**. Now, ask yourself after the **k’th** pirate has taken his share, what is the remanant number of coins ? This is **( 12-k / 12 )** of what was originally there. * [ Why ? Because each pirate takes k/12 of the coins, remember ? ]* Now, could you try taking things up from here…by yourself ?

Let’s understand the next thing the problem is trying to focus on. * “Each pirate receives a whole number of coins”* Now, this should actually help us conclude

**x. ( (11.10.9.8.7.6.5.4.3.2.1) / 12 ) is supposed to be an integer**. Since this actually implies divisibility. Cancellation of terms leads us to :

**x. ( (11.5.1.7.1.5.1.1.1.1) / ( 12.6.2.12.2.12.3.4.6.12 ) )**Can you try and approach the solution by yourself now ?

Now, this tells us the intuition of the problem. We make sure that the quotient should be an integer ! Also, recall that the 12’th pirate definitely takes the entirety of what is left, practically unity since it is exactly divisible. * *

So basically, we just realized that the denominator is entirely multiplied out…cancelled ! **And since we know that the denominator cancels out, the number of gold coins received by the 12th pirate is just going to be the product of the numerators !!!** That evaluates to : * 11.5.7.5 = 1925* And that completes our solution !

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#### Math Olympiad Program

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