1. ## arithmetics modulo

Hello,
I am a student in discrete mathematics and I have trouble solving this problem.
Any help track is welcome!
Here is the problem:

A boat sinks in the ocean and the 5 survivors each jump in a different lifeboat.They meet on the deserted island in the distance.The next morning, the first shipwrecker on the beach finds a huge pile of 'oranges.Il decides to separate them into 5 equal parts.Apres the distribution, it remains one, that He throws to the sea. He leaves to discover the island with his packet of oranges.

Shortly after the second shipwreck finds the pile of remaining oranges.Seing the first to arrive, he decides to separate it into 5 equal parts.After the distribution, there remains one that he throws into the sea, then he s The 3.4 and 5 th shipwrecks do just like the others each in their turn.

What is the minimum number of oranges that was on the beach?

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2. ## Re: arithmetics modulo

What is the minimum number for the 5th individual?

3. ## Re: arithmetics modulo

Let

$\displaystyle x=$ the total number of oranges

and suppose the five shipwrecked persons take from the pile $\displaystyle a,b,c,d,e$ oranges

Solve the five equations in six unknowns

$\displaystyle x=5a+1$

$\displaystyle 4a=5b+1$

$\displaystyle 4b=5c+1$

$\displaystyle 4c=5d+1$

$\displaystyle 4d=5e+1$

Rewrite

$\displaystyle x+4=5(a+1)$
$\displaystyle 4(a+1)=5(b+1)$
$\displaystyle 4(b+1)=5(c+1)$
$\displaystyle 4(c+1)=5(d+1)$
$\displaystyle 4(d+1)=5(e+1)$

Multiply

$\displaystyle 4^4(x+4)=5^5(e+1)$