**Solution:**

Let's assume that the number of sides of the polygon is 'n'.

Given that the smallest angle is 100 degrees and the common difference between the angles is 10 degrees.

We know that the sum of the internal angles of a polygon is given by the formula: (n-2) * 180 degrees.

Since the internal angles are in arithmetic progression (AP), the sum of the angles can also be expressed as the sum of an AP.

The sum of an AP is given by the formula: (n/2) * (2a + (n-1)d), where 'a' is the first term and 'd' is the common difference of the AP.

In this case, the first term 'a' is 100 degrees and the common difference 'd' is 10 degrees.

So, the sum of the internal angles of the polygon can be expressed as: (n/2) * (2 * 100 + (n-1) * 10).

Equating this expression to the sum of the internal angles of the polygon using the formula (n-2) * 180, we get:

(n/2) * (2 * 100 + (n-1) * 10) = (n-2) * 180.

Simplifying this equation, we get: 200n + 10n^2 - 10n = 360n - 720.

Rearranging the equation, we get: 10n^2 - 370n + 720 = 0.

Now, we can solve this quadratic equation to find the value of 'n'.

Using factoring or the quadratic formula, we find that the solutions to this equation are n = 36 and n = 20.

Since the number of sides cannot be negative or zero, we discard the solution n = 20.

Therefore, the number of sides of the polygon is 36.

Hence, the polygon has 36 sides.

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