The number between 4000 and 5000 that is divisible by each of 12, 18, 21 and 32 is

Options

A. 4023
B. 4032
C. 4302
D. 4203

Answer

To find the number between 4000 and 5000 that is divisible by 12, 18, 21, and 32, we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of the numbers 12, 18, 21, and 32.

**Finding the prime factorization:**
- 12 = 2^2 × 3^1
- 18 = 2^1 × 3^2
- 21 = 3^1 × 7^1
- 32 = 2^5

**Taking the highest power of each prime factor:**
- For 2: the highest power is 2^5 (from 32)
- For 3: the highest power is 3^2 (from 18)
- For 7: the highest power is 7^1 (from 21)

**Calculating the LCM:**
LCM = 2^5 × 3^2 × 7^1
= 32 × 9 × 7
= 32 × 63
= 2016

### Step 2: Find the multiples of the LCM that lie between 4000 and 5000.

**Finding the first multiple of 2016 greater than 4000:**
To find the smallest integer \( n \) such that \( 2016n > 4000 \):
- \( n = \lceil \frac{4000}{2016} \rceil \)
- \( n = \lceil 1.980 \rceil = 2 \)

**Calculating the multiple:**
- \( 2016 \times 2 = 4032 \)

### Step 3: Check if the next multiple of 2016 is still within the range.

**Finding the next multiple:**
- \( 2016 \times 3 = 6048 \)

Since 6048 is greater than 5000, we do not need to consider it.

### Conclusion:
The number between 4000 and 5000 that is divisible by 12, 18, 21, and 32 is **4032**.

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