The Old Egg Puzzle

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
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