How many numbers are there from 5 to 125 which are divisible by both 3 and 7?

Options

A. 6
B. 4
C. 2
D. 5

Answer

To solve the problem of finding how many numbers from 5 to 125 are divisible by both 3 and 7, we can follow these steps:

### Step 1: Find the LCM of 3 and 7
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both.
- The prime factorization of 3 is \(3^1\).
- The prime factorization of 7 is \(7^1\).
- Therefore, the LCM of 3 and 7 is \(3^1 \times 7^1 = 21\).

**Hint:** To find the LCM, consider the prime factors of the numbers and take the highest power of each prime.

### Step 2: Identify the range
We need to find the multiples of 21 that lie between 5 and 125.

**Hint:** Always define the range clearly before looking for multiples.

### Step 3: Find the smallest multiple of 21 that is greater than or equal to 5
The first multiple of 21 is 21 itself, which is greater than 5.

**Hint:** Start from the smallest number in your range and find the first multiple of the LCM.

### Step 4: Find the largest multiple of 21 that is less than or equal to 125
To find the largest multiple of 21 within the range, we can divide 125 by 21:
\[ 125 \div 21 \approx 5.95 \]
Taking the integer part, we have 5. Therefore, the largest multiple is:
\[ 21 \times 5 = 105 \]

**Hint:** Use division to find how many times the LCM fits into the upper limit of your range.

### Step 5: List the multiples of 21 between 5 and 125
The multiples of 21 from 21 to 105 are:
- \(21 \times 1 = 21\)
- \(21 \times 2 = 42\)
- \(21 \times 3 = 63\)
- \(21 \times 4 = 84\)
- \(21 \times 5 = 105\)

This gives us the multiples: 21, 42, 63, 84, and 105.

**Hint:** Listing out the multiples helps to visualize the solution and confirm your calculations.

### Step 6: Count the multiples
From the list, we see there are 5 multiples of 21 between 5 and 125.

**Hint:** Always double-check your counting to ensure accuracy.

### Conclusion
The total number of numbers from 5 to 125 that are divisible by both 3 and 7 is **5**.

**Final Answer:** Option D is correct.