Explanation:

To determine the least value of \(x + y\) for the number 56x34y4 to be divisible by 72, consider:

- Divisibility by 72 requires divisibility by both 8 and 9.

- Divisibility by 8: The last three digits (4y4) must be divisible by 8. Possible values for y are 0, 2, 4, 6, 8.

- Divisibility by 9: The sum of all digits \(5 + 6 + x + 3 + 4 + y + 4 = x + y + 22\) must be divisible by 9.

Calculate for minimum \(x + y\):

- For y = 0, x + 22 must be divisible by 9. So, \(x + 0 + 22 = x + 22\).

- \(22 \equiv 4 \pmod{9}\). Hence, \(x \equiv 5 \pmod{9}\), x = 5. Giving \(x + y = 5\).

- Checking divibility by 8: Last three, 404 is divisible by 8.

Hence, the least value of \(x + y\) is 5. Thus, option 3 is correct.

- Option 1: 8 - Incorrect, higher than 5.

- Option 2: 12 - Incorrect, higher than 5.

- Option 3: 5 - Correct Answer.

- Option 4: 14 - Incorrect, higher than 5.