The number of ordered pairs (x, y) of integers satisfying `x^3 + y^3 = 65` is

Options

A. 0
B. 2
C. 4
D. 6

Answer

To find the number of ordered pairs \((x, y)\) of integers that satisfy the equation \(x^3 + y^3 = 65\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^3 + y^3 = 65 \]
We can also use the identity for the sum of cubes:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
However, for this problem, we will directly check integer values for \(x\) and \(y\).

### Step 2: Determine possible values for \(x\)
Since \(x^3\) and \(y^3\) are both non-negative when \(x\) and \(y\) are integers, we can find the maximum possible value for \(x\) by calculating the cube root of 65:
\[ x \leq \sqrt[3]{65} \approx 4.02 \]
Thus, \(x\) can take integer values from \(-4\) to \(4\).

### Step 3: Check integer values for \(x\)
We will check each integer value of \(x\) from \(-4\) to \(4\) and see if \(y\) is an integer.

1. **For \(x = -4\)**:
\[ (-4)^3 + y^3 = 65 \implies -64 + y^3 = 65 \implies y^3 = 129 \quad \text{(not an integer)} \]

2. **For \(x = -3\)**:
\[ (-3)^3 + y^3 = 65 \implies -27 + y^3 = 65 \implies y^3 = 92 \quad \text{(not an integer)} \]

3. **For \(x = -2\)**:
\[ (-2)^3 + y^3 = 65 \implies -8 + y^3 = 65 \implies y^3 = 73 \quad \text{(not an integer)} \]

4. **For \(x = -1\)**:
\[ (-1)^3 + y^3 = 65 \implies -1 + y^3 = 65 \implies y^3 = 66 \quad \text{(not an integer)} \]

5. **For \(x = 0\)**:
\[ 0^3 + y^3 = 65 \implies y^3 = 65 \quad \text{(not an integer)} \]

6. **For \(x = 1\)**:
\[ 1^3 + y^3 = 65 \implies 1 + y^3 = 65 \implies y^3 = 64 \implies y = 4 \quad \text{(valid)} \]

7. **For \(x = 2\)**:
\[ 2^3 + y^3 = 65 \implies 8 + y^3 = 65 \implies y^3 = 57 \quad \text{(not an integer)} \]

8. **For \(x = 3\)**:
\[ 3^3 + y^3 = 65 \implies 27 + y^3 = 65 \implies y^3 = 38 \quad \text{(not an integer)} \]

9. **For \(x = 4\)**:
\[ 4^3 + y^3 = 65 \implies 64 + y^3 = 65 \implies y^3 = 1 \implies y = 1 \quad \text{(valid)} \]

### Step 4: List valid pairs
From our calculations, we found two valid pairs:
1. \((1, 4)\)
2. \((4, 1)\)

### Step 5: Count ordered pairs
The ordered pairs satisfying the equation are:
- \((1, 4)\)
- \((4, 1)\)

Thus, the total number of ordered pairs \((x, y)\) of integers satisfying \(x^3 + y^3 = 65\) is:
\[ \boxed{2} \]