How many two digit numbers are divisible by 3 but not by 7?

Options

A. 13
B. 26
C. 30
D. 47

Answer

To find how many two-digit numbers are divisible by 3 but not by 7, we can follow these steps:

### Step 1: Identify the range of two-digit numbers
The two-digit numbers range from 10 to 99.

### Step 2: Find the two-digit numbers divisible by 3
To find the two-digit numbers divisible by 3, we need to identify the smallest and largest two-digit numbers that are divisible by 3.

- The smallest two-digit number divisible by 3 is 12 (since 10 and 11 are not divisible by 3).
- The largest two-digit number divisible by 3 is 99 (since 99 is divisible by 3).

### Step 3: Use the formula for an arithmetic sequence
The two-digit numbers divisible by 3 form an arithmetic sequence where:
- First term (a) = 12
- Last term (l) = 99
- Common difference (d) = 3

The number of terms (n) in an arithmetic sequence can be calculated using the formula:
\[ n = \frac{l - a}{d} + 1 \]

Substituting the values:
\[ n = \frac{99 - 12}{3} + 1 \]
\[ n = \frac{87}{3} + 1 \]
\[ n = 29 + 1 \]
\[ n = 30 \]

So, there are 30 two-digit numbers that are divisible by 3.

### Step 4: Find the two-digit numbers divisible by both 3 and 7
Next, we need to find the two-digit numbers that are divisible by both 3 and 7. The least common multiple (LCM) of 3 and 7 is 21.

- The smallest two-digit number divisible by 21 is 21.
- The largest two-digit number divisible by 21 is 84.

### Step 5: Count the two-digit numbers divisible by 21
Using the same arithmetic sequence formula:
- First term (a) = 21
- Last term (l) = 84
- Common difference (d) = 21

Calculating the number of terms (n):
\[ n = \frac{l - a}{d} + 1 \]
\[ n = \frac{84 - 21}{21} + 1 \]
\[ n = \frac{63}{21} + 1 \]
\[ n = 3 + 1 \]
\[ n = 4 \]

So, there are 4 two-digit numbers that are divisible by both 3 and 7 (which are 21, 42, 63, and 84).

### Step 6: Calculate the final count
To find the two-digit numbers that are divisible by 3 but not by 7, we subtract the count of numbers that are divisible by both 3 and 7 from the count of numbers that are divisible by 3:
\[ \text{Total} = 30 - 4 = 26 \]

Thus, the final answer is that there are **26 two-digit numbers that are divisible by 3 but not by 7**.