https://r2.erweima.ai/mathbot/image/17259227031993q908zyd.jpg

The problem provided in the image states:

\[
\overline{345n} = 4^\circ
\]

where the notation \(\overline{345n}\) represents a 4-digit number, and we are asked to calculate the sum of the values that \(n\) can take.

### Interpretation:

- \(\overline{345n}\) is a 4-digit number in the form \(345n\), where \(n\) represents a digit (i.e., an integer between 0 and 9).
- The symbol \(4^\circ\) suggests that the number is divisible by 4. Thus, the number \(345n\) must be divisible by 4.

### Rule for divisibility by 4:
For a number to be divisible by 4, the last two digits must form a number divisible by 4. Therefore, we need to check when the last two digits of \(345n\), that is \(45n\), are divisible by 4.

### Values of \(n\) that satisfy the divisibility condition:

We need to check \(45n\) for \(n = 0, 1, \ldots, 9\) and find which values are divisible by 4.

- \(450 \div 4 = 112.5\) (not divisible)
- \(451 \div 4 = 112.75\) (not divisible)
- \(452 \div 4 = 113\) (divisible)
- \(453 \div 4 = 113.25\) (not divisible)
- \(454 \div 4 = 113.5\) (not divisible)
- \(455 \div 4 = 113.75\) (not divisible)
- \(456 \div 4 = 114\) (divisible)
- \(457 \div 4 = 114.25\) (not divisible)
- \(458 \div 4 = 114.5\) (not divisible)
- \(459 \div 4 = 114.75\) (not divisible)

Thus, \(n = 2\) and \(n = 6\) are the valid values.

### Sum of the values of \(n\):
\[
2 + 6 = 8
\]

The sum of the values that \(n\) can take is \(8\).

---

Would you like more details or explanations on any part of the solution? 

Here are some additional related questions:
1. How can we generalize divisibility rules for larger numbers?
2. What is the rule for divisibility by 4, and how is it applied to various numbers?
3. How would the result change if we were checking for divisibility by 8 instead of 4?
4. Can you provide examples of other 4-digit numbers divisible by 4?
5. What is the importance of divisibility rules in modular arithmetic?

**Tip**: Divisibility rules help simplify calculations in modular arithmetic, making it easier to test factors and remainders in larger numbers.

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

Learn how to determine the sum of values that 'n' can take in the number 345n for it to be divisible by 4. This problem covers divisibility rules and is suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation