How many numbers are there between 4000 and 6000 which are exactly divisible by 32, 40, 48 and 60?

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find how many numbers between 4000 and 6000 are exactly divisible by 32, 40, 48, and 60, we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the LCM of the numbers 32, 40, 48, and 60.

**Prime Factorization:**
- 32 = 2^5
- 40 = 2^3 × 5^1
- 48 = 2^4 × 3^1
- 60 = 2^2 × 3^1 × 5^1

**Taking the highest power of each prime:**
- For 2: the highest power is 2^5 (from 32)
- For 3: the highest power is 3^1 (from 48 and 60)
- For 5: the highest power is 5^1 (from 40 and 60)

**Calculating LCM:**
LCM = 2^5 × 3^1 × 5^1 = 32 × 3 × 5 = 480

### Step 2: Find the Multiples of LCM between 4000 and 6000
Next, we will find the multiples of 480 that lie between 4000 and 6000.

**Finding the smallest multiple of 480 greater than 4000:**
To find the smallest multiple of 480 greater than 4000, we divide 4000 by 480 and round up to the nearest whole number:
4000 ÷ 480 ≈ 8.33
Rounding up gives us 9.

Now, we calculate:
9 × 480 = 4320

**Finding the largest multiple of 480 less than 6000:**
To find the largest multiple of 480 less than 6000, we divide 6000 by 480:
6000 ÷ 480 ≈ 12.5
Rounding down gives us 12.

Now, we calculate:
12 × 480 = 5760

### Step 3: List the Multiples of 480 between 4000 and 6000
Now we can list the multiples of 480 from 9 to 12:
- 9 × 480 = 4320
- 10 × 480 = 4800
- 11 × 480 = 5280
- 12 × 480 = 5760

### Step 4: Count the Numbers
The numbers between 4000 and 6000 that are multiples of 480 are:
- 4320
- 4800
- 5280
- 5760

Thus, there are **4 numbers** between 4000 and 6000 that are exactly divisible by 32, 40, 48, and 60.

### Final Answer
**There are 4 numbers between 4000 and 6000 that are exactly divisible by 32, 40, 48, and 60.**

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