If eleven A.M. s are inserted between 28 and 10, then find the number of integral A.M. s.

Explanation

To solve the problem of finding the number of integral arithmetic means (A.M.s) between 28 and 10 when 11 A.M.s are inserted, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to insert 11 arithmetic means between the numbers 28 and 10. This means we will have a sequence that starts with 28, followed by 11 A.M.s, and ends with 10.

2. **Identifying the Terms**:
The sequence can be represented as:
\( a_1 = 28, a_2, a_3, ..., a_{12} = 10 \)
Here, \( a_1 \) is the first term and \( a_{12} \) is the last term.

3. **Finding the Total Number of Terms**:
The total number of terms in the sequence is 13 (11 A.M.s + 2 endpoints).

4. **Using the Arithmetic Sequence Formula**:
The formula for the nth term of an arithmetic sequence is given by:
\[ a_n = a + (n-1)d \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

5. **Setting Up the Equation**:
For our sequence, we know:
\[ a_{12} = a_1 + (12-1)d = 28 + 11d = 10 \]
Rearranging gives:
\[ 11d = 10 - 28 \]
\[ 11d = -18 \]
\[ d = -\frac{18}{11} \]

6. **Finding the A.M.s**:
The A.M.s can be calculated using the common difference \( d \):
\[ a_n = 28 + (n-1) \left(-\frac{18}{11}\right) \]
for \( n = 2, 3, \ldots, 12 \).

7. **Finding Integral A.M.s**:
For the A.M.s to be integral, the expression \( 28 + (n-1) \left(-\frac{18}{11}\right) \) must yield an integer. This means that \( (n-1) \) must be a multiple of 11.

8. **Identifying Valid Values for \( n \)**:
The values of \( n-1 \) can be \( 0, 11, 22, \ldots \) but since \( n \) ranges from 2 to 12, the only valid multiples of 11 within this range are:
- \( n-1 = 0 \) (i.e., \( n = 2 \))
- \( n-1 = 11 \) (i.e., \( n = 12 \))

9. **Calculating the Integral A.M.s**:
The valid integral A.M.s can be calculated for \( n = 2, 4, 6, 8, 10 \), which gives us:
- For \( n = 2 \): \( 28 - \frac{18}{11} = 27.64 \) (not integral)
- For \( n = 4 \): \( 28 - \frac{54}{11} = 26.82 \) (not integral)
- For \( n = 6 \): \( 28 - \frac{90}{11} = 26.00 \) (integral)
- For \( n = 8 \): \( 28 - \frac{126}{11} = 25.18 \) (not integral)
- For \( n = 10 \): \( 28 - \frac{162}{11} = 24.36 \) (not integral)

10. **Conclusion**:
The only integral A.M.s are those calculated for even values of \( n \) which are \( 2, 4, 6, 8, 10 \). Thus, there are a total of **5 integral A.M.s**.