Find the largest possible number of 5 digits which is exactly divisible by 32, 36 and 40.

Answer

To find the largest possible 5-digit number that is exactly divisible by 32, 36, and 40, we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of the numbers 32, 36, and 40.

1. **Prime Factorization**:
- 32 = 2^5
- 36 = 2^2 × 3^2
- 40 = 2^3 × 5^1

2. **Identify the highest powers of all prime factors**:
- For 2: The highest power is 2^5 (from 32).
- For 3: The highest power is 3^2 (from 36).
- For 5: The highest power is 5^1 (from 40).

3. **Calculate the LCM**:
\[ \text{LCM} = 2^5 × 3^2 × 5^1 = 32 × 9 × 5 \]
\[ = 32 × 45 = 1440 \]

### Step 2: Find the largest 5-digit number.

The largest 5-digit number is 99999.

### Step 3: Divide the largest 5-digit number by the LCM to find the largest multiple.

1. **Divide 99999 by 1440**:
\[ 99999 ÷ 1440 \approx 69.444 \]

2. **Take the integer part**:
The largest integer is 69.

### Step 4: Multiply the LCM by this integer.

1. **Calculate the largest multiple**:
\[ 69 × 1440 = 99360 \]

### Conclusion:
The largest possible 5-digit number that is exactly divisible by 32, 36, and 40 is **99360**.

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