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 62 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?

To find out the number of gold coins the merchant has, we can solve the given problem using algebra.

Let's assume that the merchant has x gold coins. According to the problem, the merchant divides the coins into two unequal numbers.

Let's say the merchant divides the coins into two numbers, a and b, such that a > b.

According to the given information, we can form the following equation:

62(a - b) = a^2 - b^2

To solve this equation, let's simplify the equation using the difference of squares identity, (a^2 - b^2) = (a + b)(a - b):

62(a - b) = (a + b)(a - b)

Now, we can cancel out (a - b) from both sides:

62 = a + b

Since we know that a > b, we can deduce that a + b is the total number of gold coins the merchant has. Therefore, a + b = x.

Now, let's consider the answer options and substitute them into the equation a + b = x:

A) 82: 82 is not divisible by 62, so this option can be eliminated. B) 62: If we substitute a = 62 and b = 0 into the equation, we get 62 + 0 = 62, which satisfies the condition. This option is a potential answer. C) 42: 42 is not divisible by 62, so this option can be eliminated. D) 22: 22 is not divisible by 62, so this option can be eliminated.

Thus, the correct answer is B) 62.

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