What is the greatest four digit number which when divided by 10, 15, 21 and 28 leaves remainders 4, 9, 15 and 22, respectively?

Options

A. 9654
B. 9666
C. 9664
D. 9864

Answer

To find the greatest four-digit number that leaves specific remainders when divided by 10, 15, 21, and 28, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( x \) such that:
- \( x \equiv 4 \mod 10 \)
- \( x \equiv 9 \mod 15 \)
- \( x \equiv 15 \mod 21 \)
- \( x \equiv 22 \mod 28 \)

### Step 2: Set Up the Equations
From the information given, we can rewrite the equations:
1. \( x = 10k + 4 \) for some integer \( k \)
2. \( x = 15m + 9 \) for some integer \( m \)
3. \( x = 21n + 15 \) for some integer \( n \)
4. \( x = 28p + 22 \) for some integer \( p \)

### Step 3: Find the LCM of the Divisors
Next, we calculate the least common multiple (LCM) of the divisors 10, 15, 21, and 28.

- **Prime Factorization:**
- \( 10 = 2 \times 5 \)
- \( 15 = 3 \times 5 \)
- \( 21 = 3 \times 7 \)
- \( 28 = 2^2 \times 7 \)

- **LCM Calculation:**
- Take the highest power of each prime:
- \( 2^2 \) from 28
- \( 3^1 \) from 15 or 21
- \( 5^1 \) from 10 or 15
- \( 7^1 \) from 21 or 28
- Thus, \( \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1 = 420 \)

### Step 4: Adjust the Remainders
To find the number \( x \), we need to express it in terms of the LCM:
- Since \( x \equiv 4 \mod 10 \), we can express \( x \) as \( 10k + 4 \).
- To satisfy all conditions, we can adjust \( x \) by subtracting the remainders from the LCM:
- \( x = 420n - r \) where \( r \) is the remainder we need to adjust.

The remainders from the equations can be transformed:
- From \( 10 \): \( 4 \) means \( 10 - 4 = 6 \)
- From \( 15 \): \( 9 \) means \( 15 - 9 = 6 \)
- From \( 21 \): \( 15 \) means \( 21 - 15 = 6 \)
- From \( 28 \): \( 22 \) means \( 28 - 22 = 6 \)

### Step 5: Find the Greatest Four-Digit Number
The greatest four-digit number is 9999. We need to find the largest \( n \) such that:
\[ 420n - 6 \leq 9999 \]
\[ 420n \leq 10005 \]
\[ n \leq \frac{10005}{420} \approx 23.8 \]
Thus, the largest integer \( n \) is 23.

### Step 6: Calculate the Number
Now we calculate:
\[ x = 420 \times 23 - 6 \]
\[ x = 9660 - 6 \]
\[ x = 9654 \]

### Conclusion
The greatest four-digit number that satisfies the conditions is **9654**.

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