The Old Egg Puzzle
Number Theory
Level
5
A woman with a HUGE basket of eggs finds that
- If she removes 541 eggs at a time, there are 12 eggs left over.
- If she removes 863 eggs at a time, there are 26 eggs left over.
- If she removes 1223 eggs at a time, there are 34 eggs left over.
- If she removes 1447 eggs at a time, there are 59 eggs left over.
What is the minimum number of eggs she has?
Trivia: 541 is the 100th Prime, 863 is the 150th, 1223 is the 200th and 1447 is the 229th. I do not think this information is relevant, though
The answer is 494036022101.
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The "useless" trivia is the key! As we know that 541, 863, 1223 and 1447 are positive prime numbers, we can use the Chinese Remainder Theorem.
The Theorem can be run on Wolfram Alpha using this form: ChineseRemainder[{r1,r2,…},{m1,m2,…}] Where rn are the remainders and mn are the numbers of eggs removed.
So the form to be run on Wolfram Aplha is this:
ChineseRemainder[{12,26,34,59},{541,863,1223,1447}]