The number of ways to select 2 numbers from {0, 1, 2, 3, 4} such that the sum of the squares of the selected numbers is divisible by 5 are (repetition of digits is allowed).

Options

A. 13
B. 11
C. 5
D. 9

Answer

To solve the problem of selecting 2 numbers from the set {0, 1, 2, 3, 4} such that the sum of their squares is divisible by 5 (with repetition allowed), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Set of Numbers**:
We have the numbers {0, 1, 2, 3, 4}.

2. **Calculate the Squares Modulo 5**:
We need to find the squares of each number in the set and then take them modulo 5:
- \(0^2 \mod 5 = 0\)
- \(1^2 \mod 5 = 1\)
- \(2^2 \mod 5 = 4\)
- \(3^2 \mod 5 = 4\)
- \(4^2 \mod 5 = 1\)

Thus, the squares modulo 5 are:
- 0: \(0\)
- 1: \(1\)
- 2: \(4\)
- 3: \(4\)
- 4: \(1\)

3. **Determine Pairs (a, b) Such That \(a^2 + b^2 \equiv 0 \mod 5\)**:
We will check combinations of the squares to see which pairs sum to a multiple of 5.

- **Case 1**: If \(a^2 \equiv 0\) (i.e., \(a = 0\)), then \(b^2\) must also be \(0\):
- Possible pair: (0, 0)

- **Case 2**: If \(a^2 \equiv 1\) (i.e., \(a = 1\) or \(a = 4\)), then \(b^2\) must be \(4\) (i.e., \(b = 2\) or \(b = 3\)):
- Possible pairs: (1, 2), (1, 3), (4, 2), (4, 3)

- **Case 3**: If \(a^2 \equiv 4\) (i.e., \(a = 2\) or \(a = 3\)), then \(b^2\) must be \(1\) (i.e., \(b = 1\) or \(b = 4\)):
- Possible pairs: (2, 1), (2, 4), (3, 1), (3, 4)

4. **List All Valid Pairs**:
From the above cases, we have the following valid pairs:
- From Case 1: (0, 0)
- From Case 2: (1, 2), (1, 3), (4, 2), (4, 3)
- From Case 3: (2, 1), (2, 4), (3, 1), (3, 4)

5. **Count the Total Valid Pairs**:
Now, we count all the valid pairs:
- (0, 0)
- (1, 2)
- (1, 3)
- (4, 2)
- (4, 3)
- (2, 1)
- (2, 4)
- (3, 1)
- (3, 4)

Total valid pairs = 9.

### Conclusion:
The total number of ways to select 2 numbers from {0, 1, 2, 3, 4} such that the sum of their squares is divisible by 5 is **9**.