Let a and b be natural numbers and let q and r be the quotient and remainder respectively when `a^2+b^2` is divided by a + b . Determine the number q and r if `q^2+ r = 2000`.

Answer

To solve the problem, we need to determine the values of \( q \) (the quotient) and \( r \) (the remainder) when \( a^2 + b^2 \) is divided by \( a + b \), given the condition \( q^2 + r = 2000 \).

### Step-by-Step Solution:

1. **Understanding the Division**:
When \( a^2 + b^2 \) is divided by \( a + b \), we can express it in the form:
\[ a^2 + b^2 = (a + b) \cdot q + r \]
where \( q \) is the quotient and \( r \) is the remainder.

2. **Using the Identity**:
We can use the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
This means:
\[ a^2 + b^2 = (a + b) \cdot q + r \]
can be rewritten as:
\[ (a + b)^2 - 2ab = (a + b) \cdot q + r \]

3. **Finding Quotient and Remainder**:
From the equation above, we can see:
- The quotient \( q = a + b \)
- The remainder \( r = -2ab \)

4. **Substituting into the Condition**:
We are given the condition:
\[ q^2 + r = 2000 \]
Substituting \( q \) and \( r \):
\[ (a + b)^2 - 2ab = 2000 \]

5. **Simplifying the Equation**:
We can rewrite the equation:
\[ (a + b)^2 - 2ab = 2000 \]
This simplifies to:
\[ a^2 + b^2 = 2000 \]

6. **Finding Suitable Values for \( a \) and \( b \)**:
We can use trial and error to find natural numbers \( a \) and \( b \) such that:
\[ a^2 + b^2 = 2000 \]
Testing pairs:
- Let \( a = 40 \) and \( b = 20 \):
\[ 40^2 + 20^2 = 1600 + 400 = 2000 \]
This works.

7. **Calculating \( q \) and \( r \)**:
Now, we can calculate \( q \) and \( r \):
- \( q = a + b = 40 + 20 = 60 \)
- \( r = -2ab = -2 \cdot 40 \cdot 20 = -1600 \)

8. **Final Values**:
Thus, we find:
\[ q = 60, \quad r = -1600 \]

### Final Answer:
The values of \( q \) and \( r \) are:
\[ q = 60, \quad r = -1600 \]