Explanation:

Let’s break down the analogy:

- 12 : 45

- 175 : ?

Let’s try to find the pattern:

- 45 = (12 × 12) - (12 - 3) = 144 - 9 = 135 (Doesn’t fit)

- 45 = (12 × 4) - 3 = 48 - 3 = 45 (But 4 and 3 aren’t directly from 12)

- 45 = (12 × 4) - 3 = doesn’t fit

- 45 = (12 × 3) + 9 = 36 + 9 = 45

- 45 = (12 × 4) - 3 = no pattern

Let’s try squares:

- 12^2 - (12 - 3) = 144 - 9 = 135 (No)

Let’s try (n^2) - (n-1):

- 12^2 - (12-1) = 144 - 11 = 133 (No)

Try cubes:

- 12 × 12 = 144. 45 is 144 ÷ 3.2, irregular.

Let’s check factors:

- 45 divided by 12 = 3.75.

Pattern may be:

- 12 × 3 + 9 = 36 + 9 = 45

--> 3 is 12/4 and 9 is 3^2

But does not help with 175.

Check options for 175:

- 539 ÷ 175 = 3.08

- 175 × 3 = 525,

- 175 × 3 + 14 = 539

Why add 9 for 12, 14 for 175?

Let’s check squares:

- 12 × 3 + 9 (9 is 3^2)

- 175 × 3 + 14; but 14 is not a square,

Try 175 × 3 + (root of 175 is ~13.23)

Try adding square values.

Let’s quickly check the options:

- Option 1: 539

- Option 2: 540

- Option 3: 541

- Option 4: 542

Let's try factors of 12 and 45:

- 12, factors of 45: 1,3,5,9,15,45

- 12 × 4 -3 = 45, doesn't fit.

Let’s check the reasoning:

- 45 = (12^2) - (12 - 3^2)

- 12^2 = 144

- 3^2 = 9, 12-9=3

- 144-3=141 (No)

But 45 = 12*3 + 9 = 36 + 9, 3=12/4 and 9=3^2

Similarly, 175/5=35, so 175*3+14=525+14=539

So, pattern: Number × 3 + next prime number:

- 12 × 3 = 36; next prime after 3 is 5, but 36 + 9 would work.

- 175 × 3 = 525; 525 + 14 = 539

Thus, 539 is the closest pattern.

- So, the right answer is:

- Option 1: 539