Let the values of a gold coin and a silver coin be g ducats and s ducats, respectively.

Since a gold coin is worth x% more than a silver coin, g = ((100 + x) / 100) × s. Hence g/s = (100 + x) / 100.

Since a silver coin is worth y% less than a gold coin, s = ((100 − y) / 100) × g. Hence g/s=100/(100 − y).

Therefore (100 + x) / 100 = 100/(100 − y).

Hence (100 + x)(100 − y) = 100 × 100 = 10 000.

Because x is a positive integer, it follows that 100 + x is a factor of 10 000 with 100 + x > 0.

Therefore we need to count the factors of 10 000 which are greater than 100.

Method 1:

In this method we just list the factors that are greater than 100, and then count them.

The factors of 10000 which are greater than 100 are 125, 200, 250, 400, 500, 625, 1000, 1250, 2000, 2500, 5000 and 10 000.

So there are 12 possible values for 100 + x, Hence there are 12possible values of x, namely, 25, 100, 150, 300, 400, 525, 900, 1150, 1900, 2400, 4900 and 9900.

Method 2:

In this method we work out the number of factors, without listing them, by considering the factorization of 10 000 into primes.

The factorization of 10 000 into primes is 2⁴ × 5⁴. Therefore 10 000 has the 25 factors 2^m × 5ⁿ for 0 ≤ 4 ≤ m and 0 ≤ n ≤ 4.

One of these factors is 100, with 10 000 = 100 × 100. The other 24 factors occur in 12 pairs.

Each pair consists of one factor greater than 100 and one factor less than 100. Hence 10 000 has 12 factors greater than 100.

Therefore there are 12 possible values of 100 + x and hence 12 possible values for x.