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Let's analyze the problem step by step. 

We are tasked with finding a three-digit number \( ABC \) such that:

1. **The sum of the digits is equal to 17**.
2. **If we swap the tens and hundreds digits, the number increases by 360**.
3. **If we swap the units and hundreds digits, the number decreases by 198**.

Let's break down each condition:

Let:
- \( A \) be the hundreds digit.
- \( B \) be the tens digit.
- \( C \) be the units digit.

Thus, the original number can be expressed as:
\[
100A + 10B + C
\]

### Step 1: Translating Each Condition

#### Condition 1: Sum of Digits
The sum of the digits is given by:
\[
A + B + C = 17
\]

#### Condition 2: Swap Tens and Hundreds
Swapping the tens and hundreds digits results in the number \( 100B + 10A + C \). According to the problem, this swap increases the number by 360:
\[
100B + 10A + C = (100A + 10B + C) + 360
\]
Simplifying, we get:
\[
100B + 10A + C - 100A - 10B - C = 360
\]
\[
90B - 90A = 360
\]
Dividing by 90:
\[
B - A = 4
\]

#### Condition 3: Swap Units and Hundreds
Swapping the units and hundreds digits results in the number \( 100C + 10B + A \). According to the problem, this swap decreases the number by 198:
\[
100C + 10B + A = (100A + 10B + C) - 198
\]
Simplifying, we get:
\[
100C + 10B + A - 100A - 10B - C = -198
\]
\[
99C - 99A = -198
\]
Dividing by 99:
\[
C - A = -2
\]

### Step 2: Solving the System of Equations
We now have a system of three equations:
1. \( A + B + C = 17 \)
2. \( B - A = 4 \)
3. \( C - A = -2 \)

Let's solve these equations systematically:

From equation (2), we find:
\[
B = A + 4
\]

From equation (3), we find:
\[
C = A - 2
\]

Substitute \( B = A + 4 \) and \( C = A - 2 \) into equation (1):
\[
A + (A + 4) + (A - 2) = 17
\]
\[
3A + 2 = 17
\]
\[
3A = 15
\]
\[
A = 5
\]

Using \( A = 5 \) in the expressions for \( B \) and \( C \):
\[
B = A + 4 = 5 + 4 = 9
\]
\[
C = A - 2 = 5 - 2 = 3
\]

### Solution
The three-digit number is:
\[
100A + 10B + C = 100 \cdot 5 + 10 \cdot 9 + 3 = 500 + 90 + 3 = 593
\]

### Verification
1. **Sum of digits**: \( 5 + 9 + 3 = 17 \) (satisfied).
2. **Swapping tens and hundreds**: \( 953 - 593 = 360 \) (satisfied).
3. **Swapping units and hundreds**: \( 395 - 593 = -198 \) (satisfied).

Thus, the number is indeed **593**.

Would you like further details on any step?

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#### Related Questions:
1. How would the solution change if the sum of the digits was different?
2. What other methods could we use to solve this problem?
3. Can we generalize this method for any three-digit number with similar conditions?
4. What happens if we swap other combinations of digits in a three-digit number?
5. How can systems of linear equations be used in real-world problems?

#### Tip:
When solving digit-based problems, always translate conditions into equations based on place values (hundreds, tens, and units) to simplify the solution.

Déterminer un nombre de trois chiffres sachant que la somme de ces chiffres est égale à 17. Si on permute le chiffre des dizaines et celui des centaines, le nombre augmente de 360. Si on permute le chiffre des unités et celui des centaines, le nombre diminue de 198.

Learn how to solve for a three-digit number with specific conditions: the sum of digits equals 17, swapping tens and hundreds increases the number by 360, and swapping units and hundreds decreases the number by 198. This solution includes algebraic concepts and systems of equations suitable for students in Grades 8-10.

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