Explanation:

Let's break down the pattern:

- 8 : 508

- If we check the relation: 8³ - 8 = 512 - 8 = 504 (Not 508), but if we try 8*8*8 - 8*8 = 512 - 64 = 448 (Not matching).

- Try 8*8*8 - 8 + 4 = 512 - 8 + 4 = 508. This works.

- ? : 2193

- Using the same pattern: x³ - x + 4 = 2193

- x³ - x = 2189

- Try x = 13: 13³ - 13 = 2197 - 13 = 2184; 2184 + 4 = 2188 (Not matching).

- Try x = 15: 15³ - 15 = 3375 - 15 = 3360; 3360 + 4 = 3364 (Not matching).

- Try x = 13, add 13³ - 13 = 2197 - 13 = 2184, 2184 + 9 = 2193; here, if second term is always (number³ - number) + number, then 13³ - 13 + 13 = 2197 - 13 + 13 = 2197.

Let’s check another approach:

- Checking 2 : 4

- If we use pattern: 2³ - 2 = 8 - 2 = 6 (Not 4).

- Maybe 2² = 4, which matches.

One more try, based on this, perhaps it is simply x³ + x.

Let's check:

8 : 512 + 8 = 520, not 508.

Try formula: number × (number² - 1)

- 8 × (8² - 1) = 8 × (64 - 1) = 8 × 63 = 504 (not matching).

None matching neatly.

Try: (n³ + n²)/2 (random guess)

- 8³ = 512, 8² = 64, 512 + 64 = 576 / 2 = 288 (not matching).

Now let's check the options directly for second pair:

- For x = 13:

Try 13³ + 13 = 2197 + 13 = 2210 (not matching), 13³ - 13 = 2197 - 13 = 2184

- For x = 17:

17³ = 4913, 4913 - 17 = 4896, 4896 + 4 = 4900

- For x = 15:

15³ = 3375; 3375 - 15 = 3360; 3360 + 4 = 3364

- For x = 19:

19³ = 6859; 6859 - 19 = 6840; 6840 + 4 = 6844

But if you try 13³ + 4 = 2197 + 4 = 2201 (Not matching).

Now, let's take the simpler case of 2 : 4; only 2² = 4.

Let's check if for 8 : 508, we have 8² = 64 (not 508).

Now, try pattern: 8 × 64 - 4 = 512 - 4 = 508

So, 8 × 64 - 4 = 508

Then next, x × x² - 4 = 2193

So, x³ = 2193 + 4 = 2197;

x³ = 2197 ? x = 13

So, the correct answer is Option 2: 13

- 8 : 508 ? 8³ - 4 = 508

- 13 : 2193 ? 13³ - 4 = 2193

- 2 : 4 ? 2³ - 4 = 4

Explanation:

- The pattern is: n³ - 4 = given value.

- 8³ - 4 = 512 - 4 = 508

- 13³ - 4 = 2197 - 4 = 2193

- 2³ - 4 = 8 - 4 = 4

- Therefore, option 2 (13) is correct.

- Option 1: 17 ? 4913 - 4 = 4909

- Option 2: 13 ? 2197 - 4 = 2193

- Option 3: 19 ? 6859 - 4 = 6855

- Option 4: 15 ? 3375 - 4 = 3371