Find out the least number, which, when divided by 15, 24, 32 and 45 leaves the remainders 8, 17, 25 and 38 respectively.

Options

A. 1440
B. 1450
C. 1433
D. 1500

Answer

To find the least number that, when divided by 15, 24, 32, and 45 leaves the remainders 8, 17, 25, and 38 respectively, we can follow these steps:

### Step 1: Set Up the Equations
We can express the problem using congruences:
- Let the least number be \( x \).
- From the problem, we have:
\[ x \equiv 8 \, (\text{mod} \, 15) \]
\[ x \equiv 17 \, (\text{mod} \, 24) \]
\[ x \equiv 25 \, (\text{mod} \, 32) \]
\[ x \equiv 38 \, (\text{mod} \, 45) \]

### Step 2: Convert to Standard Form
To simplify the calculations, we can convert the congruences to a standard form by subtracting the remainders from both sides:
- For the first equation: \( x - 8 \equiv 0 \, (\text{mod} \, 15) \) → \( x - 8 = 15k_1 \)
- For the second equation: \( x - 17 \equiv 0 \, (\text{mod} \, 24) \) → \( x - 17 = 24k_2 \)
- For the third equation: \( x - 25 \equiv 0 \, (\text{mod} \, 32) \) → \( x - 25 = 32k_3 \)
- For the fourth equation: \( x - 38 \equiv 0 \, (\text{mod} \, 45) \) → \( x - 38 = 45k_4 \)

### Step 3: Find the LCM of the Divisors
Next, we need to find the least common multiple (LCM) of the divisors 15, 24, 32, and 45.

- Prime factorization:
- \( 15 = 3 \times 5 \)
- \( 24 = 2^3 \times 3 \)
- \( 32 = 2^5 \)
- \( 45 = 3^2 \times 5 \)

- LCM Calculation:
- Take the highest power of each prime:
- \( 2^5 \) from 32
- \( 3^2 \) from 45
- \( 5^1 \) from 15
- Therefore,
\[ \text{LCM} = 2^5 \times 3^2 \times 5 = 32 \times 9 \times 5 = 1440 \]

### Step 4: Solve for the Least Number
Now we can express \( x \) in terms of the LCM:
\[ x = 1440k + r \]
Where \( r \) is the common remainder we need to find.

### Step 5: Find the Common Remainder
To find \( r \), we can calculate:
- From \( 15 \): \( 15 - 8 = 7 \)
- From \( 24 \): \( 24 - 17 = 7 \)
- From \( 32 \): \( 32 - 25 = 7 \)
- From \( 45 \): \( 45 - 38 = 7 \)

Thus, the common remainder \( r = 7 \).

### Step 6: Final Calculation
Now, we substitute back to find the least number:
\[ x = 1440k + 7 \]
For the least number, we take \( k = 1 \):
\[ x = 1440 \times 1 + 7 = 1447 \]

### Final Answer
The least number is **1447**.

---