Define a number K such that it is the sum of the squares of the first M natural numbers. (i.e. K = `1^(2) + 2^(2) + ... + M^(2))` where `M lt 55`. How many values of Mexist such that K is divisible by 4?

Options

A. 10
B. 11
C. 12
D. None of these

Answer

To solve the problem, we need to determine how many values of \( M \) exist such that the sum of the squares of the first \( M \) natural numbers, denoted as \( K \), is divisible by 4. The formula for the sum of squares of the first \( M \) natural numbers is given by:

\[ K = \frac{M(M + 1)(2M + 1)}{6} \]

We need to find the values of \( M \) such that \( K \) is divisible by 4, where \( M < 55 \).

### Step 1: Analyze the divisibility condition
For \( K \) to be divisible by 4, we need:

\[ \frac{M(M + 1)(2M + 1)}{6} \equiv 0 \mod 4 \]

This means that \( M(M + 1)(2M + 1) \) must be divisible by 24 (since \( 6 \times 4 = 24 \)).

### Step 2: Factor the divisibility condition
The expression \( M(M + 1)(2M + 1) \) consists of three consecutive integers \( M \), \( M + 1 \), and \( 2M + 1 \).

1. **Divisibility by 8**: Since \( M \) and \( M + 1 \) are consecutive integers, at least one of them is even. For their product to be divisible by 8, at least one of them must be a multiple of 8.

2. **Divisibility by 3**: At least one of \( M \), \( M + 1 \), or \( 2M + 1 \) must be divisible by 3.

### Step 3: Find multiples of 8
The multiples of 8 less than 55 are:
- 8, 16, 24, 32, 40, 48

These give us potential values for \( M \) as:
- If \( M \) is a multiple of 8: \( M = 8, 16, 24, 32, 40, 48 \)
- If \( M + 1 \) is a multiple of 8: \( M = 7, 15, 23, 31, 39, 47 \)

### Step 4: Check divisibility by 3
Now we check which of these values are also divisible by 3:

1. **For \( M = 8 \)**: \( 8 \mod 3 = 2 \) (not divisible)
2. **For \( M = 16 \)**: \( 16 \mod 3 = 1 \) (not divisible)
3. **For \( M = 24 \)**: \( 24 \mod 3 = 0 \) (divisible)
4. **For \( M = 32 \)**: \( 32 \mod 3 = 2 \) (not divisible)
5. **For \( M = 40 \)**: \( 40 \mod 3 = 1 \) (not divisible)
6. **For \( M = 48 \)**: \( 48 \mod 3 = 0 \) (divisible)

Now checking the values where \( M + 1 \) is a multiple of 8:
1. **For \( M = 7 \)**: \( 7 \mod 3 = 1 \) (not divisible)
2. **For \( M = 15 \)**: \( 15 \mod 3 = 0 \) (divisible)
3. **For \( M = 23 \)**: \( 23 \mod 3 = 2 \) (not divisible)
4. **For \( M = 31 \)**: \( 31 \mod 3 = 1 \) (not divisible)
5. **For \( M = 39 \)**: \( 39 \mod 3 = 0 \) (divisible)
6. **For \( M = 47 \)**: \( 47 \mod 3 = 2 \) (not divisible)

### Step 5: Compile the valid values of \( M \)
From our checks, the valid values of \( M \) that satisfy both conditions (divisible by 8 and 3) are:
- \( M = 24 \)
- \( M = 48 \)
- \( M = 15 \)
- \( M = 39 \)

### Conclusion
Thus, the values of \( M \) such that \( K \) is divisible by 4 are \( 15, 24, 39, \) and \( 48 \). Therefore, there are **4 values of \( M \)**.

### Final Answer
The number of values of \( M \) such that \( K \) is divisible by 4 is **4**.