A number which when divided by 10 leaves remainder of 9,when divided by 9 leaves remainder of 8 and when divided by 8 leaves a remainder of 7 is

Options

A. 1539
B. 5139
C. 2519
D. 9413

Answer

To solve the problem step by step, we need to find a number that satisfies the following conditions:

1. When divided by 10, it leaves a remainder of 9.
2. When divided by 9, it leaves a remainder of 8.
3. When divided by 8, it leaves a remainder of 7.

### Step 1: Understand the Remainders
From the conditions:
- If a number leaves a remainder of 9 when divided by 10, it can be expressed as:
\[ n = 10k + 9 \quad \text{(for some integer } k\text{)} \]
- If it leaves a remainder of 8 when divided by 9:
\[ n = 9m + 8 \quad \text{(for some integer } m\text{)} \]
- If it leaves a remainder of 7 when divided by 8:
\[ n = 8p + 7 \quad \text{(for some integer } p\text{)} \]

### Step 2: Set Up the Equations
From the above equations, we can see that:
- The number \( n \) can be expressed as:
\[ n + 1 \equiv 0 \mod{10} \]
\[ n + 1 \equiv 0 \mod{9} \]
\[ n + 1 \equiv 0 \mod{8} \]
This means \( n + 1 \) is a common multiple of 10, 9, and 8.

### Step 3: Find the LCM
Now, we need to find the least common multiple (LCM) of 10, 9, and 8.

- The prime factorization of each number is:
- \( 10 = 2 \times 5 \)
- \( 9 = 3^2 \)
- \( 8 = 2^3 \)

To find the LCM, we take the highest power of each prime:
- For \( 2 \): \( 2^3 \)
- For \( 3 \): \( 3^2 \)
- For \( 5 \): \( 5^1 \)

Thus, the LCM is:
\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 \]

### Step 4: Calculate the LCM
Now we calculate:
\[ 8 \times 9 = 72 \]
\[ 72 \times 5 = 360 \]

So, \( n + 1 = 360 \).

### Step 5: Find the Required Number
To find \( n \):
\[ n = 360 - 1 = 359 \]

### Conclusion
Thus, the number we are looking for is:
\[ \boxed{359} \]