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1. A rich merchant had collected many gold coins. He did not want anybody to know about them. One day, his wife asked, "How many gold coins do we have?"
After pausing a moment, he replied, "Well! If I divide the coins into two unequal numbers, then 31 times the difference between the two numbers equals the difference between the squares of the two numbers."
The wife looked puzzled. Can you help the merchant's wife by finding out how many gold coins they have?
The merchant has 31 gold coins.
It is easy to check this... Let's divide the 31 coins into two unequal numbers, say, 26 and 5. Then,
31 (26 − 5) = (26 × 26) − (5 × 5).
Food for thought:
Note that the 31 coins can be divided into any two unequal numbers, a and b, not necessarily 26 and 5, so long as a + b = 31. Isn't that interesting?
The above problem is solved in a straightforward manner if one knows the following formula:
a2 − b2 = (a + b) (a − b)
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