The number between 22000 and 23000that is divisible by each of 12,18,21 and 32 ?

Options

A. 22176
B. 22536
C. 22032
D. 22276

Answer

To find the number between 22,000 and 23,000 that is divisible by each of 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.

**Hint:** To find the LCM, we can use the prime factorization method.

1. **Prime Factorization:**
- 12 = 2^2 × 3^1
- 18 = 2^1 × 3^2
- 21 = 3^1 × 7^1
- 32 = 2^5

2. **Take 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)

3. **Calculate the LCM:**
\[ \text{LCM} = 2^5 \times 3^2 \times 7^1 = 32 \times 9 \times 7 \]
\[ = 32 \times 63 = 2016 \]

### Step 2: Find the multiples of 2016 that lie between 22,000 and 23,000.

**Hint:** To find the multiples, divide the lower and upper limits by the LCM.

1. **Calculate the lower limit:**
\[ \frac{22000}{2016} \approx 10.9 \quad \text{(round up to 11)} \]
So, the first multiple is:
\[ 2016 \times 11 = 22176 \]

2. **Calculate the upper limit:**
\[ \frac{23000}{2016} \approx 11.4 \quad \text{(round down to 11)} \]
The next multiple would be:
\[ 2016 \times 12 = 24192 \quad \text{(this is out of range)} \]

### Conclusion:
The only multiple of 2016 that lies between 22,000 and 23,000 is **22,176**.

**Final Answer:** The number between 22,000 and 23,000 that is divisible by each of 12, 18, 21, and 32 is **22,176**.

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