The digits of a positive integer n are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when n is divided by 37 ?

Answer

To solve the problem, we need to find the four-digit positive integers whose digits are four consecutive integers in decreasing order. Then, we will calculate the remainders when these integers are divided by 37 and finally sum those remainders.

### Step-by-Step Solution:

1. **Identify the Possible Integers:**
The digits of the integer \( n \) must be four consecutive integers in decreasing order. The highest possible four-digit integer that meets this criterion is 9876. The possible integers are:
- 9876 (digits: 9, 8, 7, 6)
- 8765 (digits: 8, 7, 6, 5)
- 7654 (digits: 7, 6, 5, 4)
- 6543 (digits: 6, 5, 4, 3)
- 5432 (digits: 5, 4, 3, 2)
- 4321 (digits: 4, 3, 2, 1)
- 3210 (digits: 3, 2, 1, 0)

2. **Calculate Remainders When Divided by 37:**
We will now divide each of these integers by 37 and find the remainders.

- For \( n = 9876 \):
\[ 9876 \div 37 = 266 \quad \text{(quotient)} \]
\[ 9876 = 37 \times 266 + 34 \quad \text{(remainder: 34)} \]

- For \( n = 8765 \):
\[ 8765 \div 37 = 236 \quad \text{(quotient)} \]
\[ 8765 = 37 \times 236 + 33 \quad \text{(remainder: 33)} \]

- For \( n = 7654 \):
\[ 7654 \div 37 = 206 \quad \text{(quotient)} \]
\[ 7654 = 37 \times 206 + 32 \quad \text{(remainder: 32)} \]

- For \( n = 6543 \):
\[ 6543 \div 37 = 176 \quad \text{(quotient)} \]
\[ 6543 = 37 \times 176 + 31 \quad \text{(remainder: 31)} \]

- For \( n = 5432 \):
\[ 5432 \div 37 = 146 \quad \text{(quotient)} \]
\[ 5432 = 37 \times 146 + 30 \quad \text{(remainder: 30)} \]

- For \( n = 4321 \):
\[ 4321 \div 37 = 116 \quad \text{(quotient)} \]
\[ 4321 = 37 \times 116 + 29 \quad \text{(remainder: 29)} \]

- For \( n = 3210 \):
\[ 3210 \div 37 = 86 \quad \text{(quotient)} \]
\[ 3210 = 37 \times 86 + 28 \quad \text{(remainder: 28)} \]

3. **List of Remainders:**
The remainders we found are:
- 34
- 33
- 32
- 31
- 30
- 29
- 28

4. **Sum of Remainders:**
Now, we will sum all the remainders:
\[ 34 + 33 + 32 + 31 + 30 + 29 + 28 \]
Calculating step-by-step:
- \( 34 + 33 = 67 \)
- \( 67 + 32 = 99 \)
- \( 99 + 31 = 130 \)
- \( 130 + 30 = 160 \)
- \( 160 + 29 = 189 \)
- \( 189 + 28 = 217 \)

Thus, the sum of the possible remainders is \( 217 \).

### Final Answer:
The sum of the possible remainders when \( n \) is divided by 37 is **217**.