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. 4

Answer

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

### Step 1: Calculate the squares of the numbers
First, we need to find the squares of each number in the set {0, 1, 2, 3, 4}.

- \(0^2 = 0\)
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\) (which is \(4 \mod 5\))
- \(4^2 = 16\) (which is \(1 \mod 5\))

So, the squares of the numbers modulo 5 are:
- \(0 \mod 5 = 0\)
- \(1 \mod 5 = 1\)
- \(2 \mod 5 = 4\)
- \(3 \mod 5 = 4\)
- \(4 \mod 5 = 1\)

### Step 2: Identify pairs whose squares sum to a multiple of 5
Next, we need to find pairs \((a, b)\) such that \(a^2 + b^2 \equiv 0 \mod 5\). We can analyze the possible sums of the squares:

- If \(a^2 \equiv 0\), then \(b^2\) must also be \(0\) (i.e., both are 0).
- If \(a^2 \equiv 1\), then \(b^2\) must be \(4\) (i.e., one is 1 and the other is 2 or 3).
- If \(a^2 \equiv 4\), then \(b^2\) must be \(1\) (i.e., one is 2 or 3 and the other is 1).

### Step 3: List valid combinations
Now, we can list the valid combinations based on the above conditions:

1. **Both numbers are 0**:
- (0, 0)

2. **One number is 1 and the other is 2 or 3**:
- (1, 2)
- (1, 3)
- (2, 1)
- (3, 1)

3. **One number is 2 and the other is 1**:
- (2, 1)
- (3, 1)
- (1, 2)
- (1, 3)

4. **Both numbers are 2 or 3**:
- (2, 2)
- (3, 3)

### Step 4: Count the valid pairs
Now we can count the valid pairs:
- From (0, 0): 1 way
- From (1, 2) and (1, 3): 4 ways
- From (2, 1) and (3, 1): 4 ways
- From (2, 2) and (3, 3): 2 ways

Adding these, we have:
1 (from (0,0)) + 4 (from (1,2) and (1,3)) + 4 (from (2,1) and (3,1)) + 2 (from (2,2) and (3,3)) = 11 ways.

### Final Answer
The total number of ways to select 2 numbers from the set {0, 1, 2, 3, 4} such that the sum of their squares is divisible by 5 is **11**.