The King's Coins

     The second half of a famous "Mother Goose" nursery rhyme goes something like this:

The King was in the counting-house,
Counting out his money,
The Queen was in the parlour,
Eating bread and honey,

The maid was in the garden,
Hanging out the clothes,
When down came a little bird
And snapped off her nose!

     Now, I have no particular interest in the affairs of those two ladies. As the king's Royal Mathematician, however, I do have a curiosity about how much money he was counting.

     So I went to His Highness to inquire.

     His reply to my question as to how many gold coins he had on His counting table was rather unexpected. He said:

     "If I put all my coins in stacks of 25 each, I have 15 coins left over in another, shorter stack. If I then put them in stacks of 35 each, I still have 15 left over, just as before. But if I put them in stacks of 45, there are no coins left over. It comes out just right!"

     Well, my friends, I'm not called the Royal Mathematician for nothing. I took out my quill, ink bottle and a piece of parchment, and set to work. In no time at all, I knew just how many gold coins the King had.

     To answer this problem, tell me the number of stacks of coins the King had in His first counting method.

     Remember: define your variables carefully, set up good equations, and use algebraic thinking. (And remember; there were no calculators nor computers in the King's time!)

     BONUS: How does the first half of this famous rhyme go?

NOTE: Before writing out your answer, please check our Guide lines for Writing POW Answers.

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