What is the least possible number which when divided by 24, 32 or 42 in each case it leaves the remainder 5?

Answer

To solve the problem of finding the least possible number which, when divided by 24, 32, or 42, leaves a remainder of 5, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( N \) such that:
- \( N \mod 24 = 5 \)
- \( N \mod 32 = 5 \)
- \( N \mod 42 = 5 \)

This means that \( N - 5 \) must be divisible by 24, 32, and 42.

### Step 2: Set Up the Equation
Let \( M = N - 5 \). Then, we need to find \( M \) such that:
- \( M \mod 24 = 0 \)
- \( M \mod 32 = 0 \)
- \( M \mod 42 = 0 \)

This means \( M \) is a common multiple of 24, 32, and 42.

### Step 3: Find the LCM
To find \( M \), we need to calculate the least common multiple (LCM) of 24, 32, and 42.

1. **Prime Factorization**:
- \( 24 = 2^3 \times 3^1 \)
- \( 32 = 2^5 \)
- \( 42 = 2^1 \times 3^1 \times 7^1 \)

2. **Determine the LCM**:
- For \( 2 \): The highest power is \( 2^5 \) (from 32).
- For \( 3 \): The highest power is \( 3^1 \) (from both 24 and 42).
- For \( 7 \): The highest power is \( 7^1 \) (from 42).

Thus, the LCM is:
\[ \text{LCM} = 2^5 \times 3^1 \times 7^1 \]

### Step 4: Calculate the LCM
Now we calculate the LCM:
\[ \text{LCM} = 32 \times 3 \times 7 \]
Calculating step by step:
- \( 32 \times 3 = 96 \)
- \( 96 \times 7 = 672 \)

So, \( M = 672 \).

### Step 5: Find \( N \)
Since \( N = M + 5 \):
\[ N = 672 + 5 = 677 \]

### Conclusion
The least possible number which when divided by 24, 32, or 42 leaves a remainder of 5 is \( \boxed{677} \).

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