To solve this problem, we can follow these steps:

Step 1: Calculate the sum of the interior angles of a 7-sided polygon.

The formula to calculate the sum of the interior angles of a polygon with $n$ sides is: $\text{Sum of interior angles} = 180^\circ \times (n - 2)$ For a 7-sided polygon: $\text{Sum of interior angles} = 180^\circ \times (7 - 2) = 180^\circ \times 5 = 900^\circ$

Step 2: Set up the equation for the angles.

Let the number of angles that measure 124° be $k$. Therefore, the number of angles that measure $x^\circ$ (a multiple of 10) will be $7 - k$.

The sum of the interior angles is then given by: $k \times 124^\circ + (7 - k) \times x^\circ = 900^\circ$

Step 3: Simplify the equation.

Expand and rearrange the equation: $124k + (7 - k)x = 900$ $124k + 7x - kx = 900$ $k(124 - x) + 7x = 900$

Step 4: Test for values of $k$ and solve for $x$.

Since $x$ is a multiple of 10, we will substitute potential values of $x$ and solve for $k$ to ensure $k$ is an integer.

Let's start testing for multiples of 10:

1. If $x = 120^\circ$: $k(124 - 120) + 7(120) = 900$ $4k + 840 = 900$ $4k = 60$ k = 15 \, \text{(not possible since k cannot be greater than 7)}

2. If $x = 110^\circ$: $k(124 - 110) + 7(110) = 900$ $14k + 770 = 900$ $14k = 130$ $k = \frac{130}{14} \approx 9.29 \, \text{(not an integer)}$

3. If $x = 100^\circ$: $k(124 - 100) + 7(100) = 900$ $24k + 700 = 900$ $24k = 200$ $k = \frac{200}{24} \approx 8.33 \, \text{(not an integer)}$

4. If $x = 130^\circ$: $k(124 - 130) + 7(130) = 900$ $k(-6) + 910 = 900$ $-6k = -10$ $k = \frac{10}{6} \approx 1.67 \, \text{(not an integer)}$

5. If $x = 140^\circ$: $k(124 - 140) + 7(140) = 900$ $-16k + 980 = 900$ $-16k = -80$ $k = 5 \, \text{(an integer and possible!)}$

Step 5: Verification

If $k = 5$ and $x = 140^\circ$: $5(124) + 2(140) = 620 + 280 = 900^\circ$ This satisfies the total sum of the angles.

Thus, the correct value of $x$ is $\boxed{140^\circ}$.

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Would you like any additional clarification? Here are some follow-up questions:

1. How would the calculation change for a different polygon?
2. What if $x$ was not a multiple of 10?
3. Can you generalize the method for any $n$-sided polygon?
4. How would this change if there were more or fewer angles of 124°?
5. How does the requirement for $x$ being a multiple of 10 simplify the problem?

Tip: When solving for multiple unknowns with constraints, test values systematically to avoid missing possible solutions.