
These two relatives celebrated their birthdays this year (in 2024, like every year), but this time:

- Person A has reached the age that corresponds to person B's year of birth.
- Person B has reached the age that corresponds to person A's year of birth

In addition, the year of birth is assumed to be a two-digit number, e.g.
for a person who was born in 1983, the year of birth is = 83

Person A is 16 years older than person B. 

Let’s denote the ages and years of birth for Person A and Person B to solve this.

1. Let \( x \) represent the current age of Person A in 2024.
2. Let \( y \) represent the current age of Person B in 2024.

### Given Information
- **Person A has reached the age corresponding to Person B's year of birth.**
- **Person B has reached the age corresponding to Person A's year of birth.**
- **Person A is 16 years older than Person B, so \( x = y + 16 \).**

Assuming that each person’s year of birth can be represented by the last two digits:
1. **Person A's year of birth** would then be \( 2024 - x \).
   - Since this is represented by the last two digits, we take \( (2024 - x) \mod 100 \) as the two-digit year.
2. **Person B's year of birth** would be \( 2024 - y \), and similarly, this is represented by \( (2024 - y) \mod 100 \).

### Setting up Equations
Using the problem’s conditions, we have:
1. **Person A’s age** \( x \) matches **Person B’s year of birth**:  
   \[ x = (2024 - y) \mod 100 \]

2. **Person B’s age** \( y \) matches **Person A’s year of birth**:  
   \[ y = (2024 - x) \mod 100 \]

Using \( x = y + 16 \), we can solve this system of equations. 

Let me calculate these steps to find the solution.It appears that this system of equations is complex to solve directly in this format, especially due to the modular constraint. I’ll take an alternative approach by exploring possible values for \( x \) and \( y \) manually within a reasonable range.

Let’s proceed with that to find a consistent solution for \( x \) and \( y \).The possible solutions for the ages of Person A and Person B in 2024 are:

1. **Person A is 20 years old, and Person B is 4 years old.**
2. **Person A is 70 years old, and Person B is 54 years old.**

Both pairs satisfy the conditions where each person's age corresponds to the two-digit year of birth of the other, and Person A is 16 years older than Person B.

Would you like more details on how this solution was derived, or do you have further questions?

Here are five questions for practice:
1. How can modular arithmetic help in determining year-related conditions?
2. How could this problem be adapted to account for different year gaps?
3. What other age relationships could produce interesting patterns in years?
4. How would changing the 16-year age difference affect potential solutions?
5. What would the solution be if the problem used three people instead of two?

**Tip:** Always re-check modular conditions carefully, as they can lead to unique results based on the base (in this case, 100 for the last two digits).

These two relatives celebrated their birthdays this year (in 2024, like every year), but this time:
- Person A has reached the age that corresponds to person B's year of birth.
- Person B has reached the age that corresponds to person A's year of birth
In addition, the year of birth is assumed to be a two-digit number, e.g. for a person who was born in 1983, the year of birth is = 83
Person A is 16 years older than person B.

Solve an age problem where Person A and Person B reach ages that match the two-digit birth years of each other, with a 16-year age difference. This problem demonstrates the use of modular arithmetic and systems of equations. Suitable for high school students studying algebra and modular math concepts.

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