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The problem states:

"Si \(\overline{345n} = 4^\circ\), calcula la suma de valores que puede tomar \(n\)."

Here, \(\overline{345n}\) represents a number, \(345n\), whose sum of digits or digit manipulation is implied to equal \(4^\circ\). To solve this, let's follow these steps:

1. **Interpretation**: \(\overline{345n}\) means the concatenation of digits "345" followed by the digit \(n\). It represents a 4-digit number \(345n\).
2. **Condition**: \(\overline{345n} = 4^\circ\), which likely means the result of dividing \(345n\) by 4 should be an integer, implying \(345n\) must be divisible by 4.
   
   A number is divisible by 4 if its last two digits form a number divisible by 4. Hence, we should find \(n\) such that \(45n\) (the last two digits) is divisible by 4.

**Calculations**:
- Check which values of \(n\) (from 0 to 9) make \(45n\) divisible by 4:
  - For \(n = 0\): \(450 \mod 4 = 2\) (not divisible)
  - For \(n = 1\): \(451 \mod 4 = 3\) (not divisible)
  - For \(n = 2\): \(452 \mod 4 = 0\) (divisible)
  - For \(n = 3\): \(453 \mod 4 = 1\) (not divisible)
  - For \(n = 4\): \(454 \mod 4 = 2\) (not divisible)
  - For \(n = 5\): \(455 \mod 4 = 3\) (not divisible)
  - For \(n = 6\): \(456 \mod 4 = 0\) (divisible)
  - For \(n = 7\): \(457 \mod 4 = 1\) (not divisible)
  - For \(n = 8\): \(458 \mod 4 = 2\) (not divisible)
  - For \(n = 9\): \(459 \mod 4 = 3\) (not divisible)

**Valid values**: \(n = 2\) and \(n = 6\).

**Sum of values**: \(2 + 6 = 8\).

The sum of values that \(n\) can take is \(\boxed{8}\).

Would you like further details or have any questions?

Here are some related questions:
1. How do divisibility rules apply to different numbers?
2. What are other common methods to determine divisibility?
3. How can we generalize this approach for larger numbers?
4. What if the equation had a different modulus condition?
5. How can this problem be solved using programming?

**Tip**: For divisibility by 4, always check the last two digits of the number.

Si 345n = 4°, calcula la suma de valores que puede tomar «n».

Learn how to solve a divisibility problem where you calculate the sum of values for n in the expression 345n = 4°. Follow a detailed step-by-step approach using divisibility rules for numbers divisible by 4. Suitable for students in Grades 6-8.

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