Asked by Evan and Max
?^3 + ?^5 = 40
Answers
Answered by
Evan and Max
?^2 + ?^3 = 44
Answered by
Evan and Max
?^3 + ?^2 = 17
Answered by
GPT-4o mini
To find the integer solutions for the equation \( x^3 + y^5 = 40 \), where \( x \) and \( y \) are integers, we can start by testing various values for \( y \) and calculating the corresponding \( x \).
1. **For \( y = 0 \)**:
\[
x^3 + 0^5 = 40 \implies x^3 = 40 \implies x \approx 3.4 \quad \text{(no integer solution)}
\]
2. **For \( y = 1 \)**:
\[
x^3 + 1^5 = 40 \implies x^3 + 1 = 40 \implies x^3 = 39 \implies x \approx 3.4 \quad \text{(no integer solution)}
\]
3. **For \( y = 2 \)**:
\[
x^3 + 2^5 = 40 \implies x^3 + 32 = 40 \implies x^3 = 8 \implies x = 2 \quad \text{(solution: } (2, 2) \text{)}
\]
4. **For \( y = 3 \)**:
\[
x^3 + 3^5 = 40 \implies x^3 + 243 = 40 \implies x^3 = -203 \quad \text{(no integer solution)}
\]
5. **For \( y = -1 \)**:
\[
x^3 + (-1)^5 = 40 \implies x^3 - 1 = 40 \implies x^3 = 41 \implies x \approx 3.4 \quad \text{(no integer solution)}
\]
6. **For \( y = -2 \)**:
\[
x^3 + (-2)^5 = 40 \implies x^3 - 32 = 40 \implies x^3 = 72 \implies x \approx 4.16 \quad \text{(no integer solution)}
\]
Continuing similarly for negative values of \( y \) and larger positive values will yield no further integer solutions. After checking the values sequentially, we conclude that the only integer solution to the equation \( x^3 + y^5 = 40 \) is:
\[
\boxed{(2, 2)}
\]
1. **For \( y = 0 \)**:
\[
x^3 + 0^5 = 40 \implies x^3 = 40 \implies x \approx 3.4 \quad \text{(no integer solution)}
\]
2. **For \( y = 1 \)**:
\[
x^3 + 1^5 = 40 \implies x^3 + 1 = 40 \implies x^3 = 39 \implies x \approx 3.4 \quad \text{(no integer solution)}
\]
3. **For \( y = 2 \)**:
\[
x^3 + 2^5 = 40 \implies x^3 + 32 = 40 \implies x^3 = 8 \implies x = 2 \quad \text{(solution: } (2, 2) \text{)}
\]
4. **For \( y = 3 \)**:
\[
x^3 + 3^5 = 40 \implies x^3 + 243 = 40 \implies x^3 = -203 \quad \text{(no integer solution)}
\]
5. **For \( y = -1 \)**:
\[
x^3 + (-1)^5 = 40 \implies x^3 - 1 = 40 \implies x^3 = 41 \implies x \approx 3.4 \quad \text{(no integer solution)}
\]
6. **For \( y = -2 \)**:
\[
x^3 + (-2)^5 = 40 \implies x^3 - 32 = 40 \implies x^3 = 72 \implies x \approx 4.16 \quad \text{(no integer solution)}
\]
Continuing similarly for negative values of \( y \) and larger positive values will yield no further integer solutions. After checking the values sequentially, we conclude that the only integer solution to the equation \( x^3 + y^5 = 40 \) is:
\[
\boxed{(2, 2)}
\]
Answered by
GPT-4o mini
To find integer solutions for the equation \( x^2 + y^3 = 44 \), we can systematically check various integer values for \( y \) and solve for \( x \).
1. **For \( y = 0 \)**:
\[
x^2 + 0^3 = 44 \implies x^2 = 44 \implies x \approx 6.63 \quad \text{(no integer solution)}
\]
2. **For \( y = 1 \)**:
\[
x^2 + 1^3 = 44 \implies x^2 + 1 = 44 \implies x^2 = 43 \implies x \approx 6.56 \quad \text{(no integer solution)}
\]
3. **For \( y = 2 \)**:
\[
x^2 + 2^3 = 44 \implies x^2 + 8 = 44 \implies x^2 = 36 \implies x = 6 \quad \text{(solution: } (6, 2) \text{)}
\]
4. **For \( y = 3 \)**:
\[
x^2 + 3^3 = 44 \implies x^2 + 27 = 44 \implies x^2 = 17 \implies x \approx 4.12 \quad \text{(no integer solution)}
\]
5. **For \( y = 4 \)**:
\[
x^2 + 4^3 = 44 \implies x^2 + 64 = 44 \implies x^2 = -20 \quad \text{(no integer solution)}
\]
Next, we will check negative values for \( y \):
6. **For \( y = -1 \)**:
\[
x^2 + (-1)^3 = 44 \implies x^2 - 1 = 44 \implies x^2 = 45 \implies x \approx 6.71 \quad \text{(no integer solution)}
\]
7. **For \( y = -2 \)**:
\[
x^2 + (-2)^3 = 44 \implies x^2 - 8 = 44 \implies x^2 = 52 \implies x \approx 7.21 \quad \text{(no integer solution)}
\]
8. **For \( y = -3 \)**:
\[
x^2 + (-3)^3 = 44 \implies x^2 - 27 = 44 \implies x^2 = 71 \implies x \approx 8.43 \quad \text{(no integer solution)}
\]
9. **For \( y = -4 \)**:
\[
x^2 + (-4)^3 = 44 \implies x^2 - 64 = 44 \implies x^2 = 108 \implies x \approx 10.39 \quad \text{(no integer solution)}
\]
After checking integer values for \( y \) from \(-4\) to \(4\), the only integer solution to the equation \( x^2 + y^3 = 44 \) is:
\[
\boxed{(6, 2)}
\]
1. **For \( y = 0 \)**:
\[
x^2 + 0^3 = 44 \implies x^2 = 44 \implies x \approx 6.63 \quad \text{(no integer solution)}
\]
2. **For \( y = 1 \)**:
\[
x^2 + 1^3 = 44 \implies x^2 + 1 = 44 \implies x^2 = 43 \implies x \approx 6.56 \quad \text{(no integer solution)}
\]
3. **For \( y = 2 \)**:
\[
x^2 + 2^3 = 44 \implies x^2 + 8 = 44 \implies x^2 = 36 \implies x = 6 \quad \text{(solution: } (6, 2) \text{)}
\]
4. **For \( y = 3 \)**:
\[
x^2 + 3^3 = 44 \implies x^2 + 27 = 44 \implies x^2 = 17 \implies x \approx 4.12 \quad \text{(no integer solution)}
\]
5. **For \( y = 4 \)**:
\[
x^2 + 4^3 = 44 \implies x^2 + 64 = 44 \implies x^2 = -20 \quad \text{(no integer solution)}
\]
Next, we will check negative values for \( y \):
6. **For \( y = -1 \)**:
\[
x^2 + (-1)^3 = 44 \implies x^2 - 1 = 44 \implies x^2 = 45 \implies x \approx 6.71 \quad \text{(no integer solution)}
\]
7. **For \( y = -2 \)**:
\[
x^2 + (-2)^3 = 44 \implies x^2 - 8 = 44 \implies x^2 = 52 \implies x \approx 7.21 \quad \text{(no integer solution)}
\]
8. **For \( y = -3 \)**:
\[
x^2 + (-3)^3 = 44 \implies x^2 - 27 = 44 \implies x^2 = 71 \implies x \approx 8.43 \quad \text{(no integer solution)}
\]
9. **For \( y = -4 \)**:
\[
x^2 + (-4)^3 = 44 \implies x^2 - 64 = 44 \implies x^2 = 108 \implies x \approx 10.39 \quad \text{(no integer solution)}
\]
After checking integer values for \( y \) from \(-4\) to \(4\), the only integer solution to the equation \( x^2 + y^3 = 44 \) is:
\[
\boxed{(6, 2)}
\]
Answered by
GPT-4o mini
To find integer solutions for the equation \( x^3 + y^2 = 17 \), we can test various integer values for \( y \) and compute the corresponding \( x \).
1. **For \( y = 0 \)**:
\[
x^3 + 0^2 = 17 \implies x^3 = 17 \implies x \approx 2.57 \quad \text{(no integer solution)}
\]
2. **For \( y = 1 \)**:
\[
x^3 + 1^2 = 17 \implies x^3 + 1 = 17 \implies x^3 = 16 \implies x = 2 \quad \text{(solution: } (2, 1) \text{)}
\]
3. **For \( y = 2 \)**:
\[
x^3 + 2^2 = 17 \implies x^3 + 4 = 17 \implies x^3 = 13 \implies x \approx 2.35 \quad \text{(no integer solution)}
\]
4. **For \( y = 3 \)**:
\[
x^3 + 3^2 = 17 \implies x^3 + 9 = 17 \implies x^3 = 8 \implies x = 2 \quad \text{(solution: } (2, 3) \text{)}
\]
5. **For \( y = 4 \)**:
\[
x^3 + 4^2 = 17 \implies x^3 + 16 = 17 \implies x^3 = 1 \implies x = 1 \quad \text{(solution: } (1, 4) \text{)}
\]
6. **For \( y = 5 \)**:
\[
x^3 + 5^2 = 17 \implies x^3 + 25 = 17 \implies x^3 = -8 \implies x = -2 \quad \text{(solution: } (-2, 5) \text{)}
\]
7. **For larger values of \( y \)** (i.e., \( y \geq 6 \)), \( y^2 \) exceeds 17, making \( x^3 \) negative, leading to no positive integer solutions.
Next, we can also check negative values for \( y \):
8. **For \( y = -1 \)**:
\[
x^3 + (-1)^2 = 17 \implies x^3 + 1 = 17 \implies x^3 = 16 \implies x = 2 \quad \text{(solution: } (2, -1) \text{)}
\]
9. **For \( y = -2 \)**:
\[
x^3 + (-2)^2 = 17 \implies x^3 + 4 = 17 \implies x^3 = 13 \implies x \approx 2.35 \quad \text{(no integer solution)}
\]
10. **For \( y = -3 \)**:
\[
x^3 + (-3)^2 = 17 \implies x^3 + 9 = 17 \implies x^3 = 8 \implies x = 2 \quad \text{(solution: } (2, -3) \text{)}
\]
11. **For \( y = -4 \)**:
\[
x^3 + (-4)^2 = 17 \implies x^3 + 16 = 17 \implies x^3 = 1 \implies x = 1 \quad \text{(solution: } (1, -4) \text{)}
\]
Thus, the integer solutions for the equation \( x^3 + y^2 = 17 \) are:
\[
\boxed{(2, 1), (2, 3), (1, 4), (-2, 5), (2, -1), (2, -3), (1, -4)}
\]
1. **For \( y = 0 \)**:
\[
x^3 + 0^2 = 17 \implies x^3 = 17 \implies x \approx 2.57 \quad \text{(no integer solution)}
\]
2. **For \( y = 1 \)**:
\[
x^3 + 1^2 = 17 \implies x^3 + 1 = 17 \implies x^3 = 16 \implies x = 2 \quad \text{(solution: } (2, 1) \text{)}
\]
3. **For \( y = 2 \)**:
\[
x^3 + 2^2 = 17 \implies x^3 + 4 = 17 \implies x^3 = 13 \implies x \approx 2.35 \quad \text{(no integer solution)}
\]
4. **For \( y = 3 \)**:
\[
x^3 + 3^2 = 17 \implies x^3 + 9 = 17 \implies x^3 = 8 \implies x = 2 \quad \text{(solution: } (2, 3) \text{)}
\]
5. **For \( y = 4 \)**:
\[
x^3 + 4^2 = 17 \implies x^3 + 16 = 17 \implies x^3 = 1 \implies x = 1 \quad \text{(solution: } (1, 4) \text{)}
\]
6. **For \( y = 5 \)**:
\[
x^3 + 5^2 = 17 \implies x^3 + 25 = 17 \implies x^3 = -8 \implies x = -2 \quad \text{(solution: } (-2, 5) \text{)}
\]
7. **For larger values of \( y \)** (i.e., \( y \geq 6 \)), \( y^2 \) exceeds 17, making \( x^3 \) negative, leading to no positive integer solutions.
Next, we can also check negative values for \( y \):
8. **For \( y = -1 \)**:
\[
x^3 + (-1)^2 = 17 \implies x^3 + 1 = 17 \implies x^3 = 16 \implies x = 2 \quad \text{(solution: } (2, -1) \text{)}
\]
9. **For \( y = -2 \)**:
\[
x^3 + (-2)^2 = 17 \implies x^3 + 4 = 17 \implies x^3 = 13 \implies x \approx 2.35 \quad \text{(no integer solution)}
\]
10. **For \( y = -3 \)**:
\[
x^3 + (-3)^2 = 17 \implies x^3 + 9 = 17 \implies x^3 = 8 \implies x = 2 \quad \text{(solution: } (2, -3) \text{)}
\]
11. **For \( y = -4 \)**:
\[
x^3 + (-4)^2 = 17 \implies x^3 + 16 = 17 \implies x^3 = 1 \implies x = 1 \quad \text{(solution: } (1, -4) \text{)}
\]
Thus, the integer solutions for the equation \( x^3 + y^2 = 17 \) are:
\[
\boxed{(2, 1), (2, 3), (1, 4), (-2, 5), (2, -1), (2, -3), (1, -4)}
\]