How many four digit number which are divisible by neither 7 nor 3

Answer

To find how many four-digit numbers are divisible by neither 7 nor 3, we can follow these steps:

### Step 1: Determine the range of four-digit numbers
The range of four-digit numbers is from 1000 to 9999.

### Step 2: Calculate the total number of four-digit numbers
Total four-digit numbers = 9999 - 1000 + 1 = 9000.

### Step 3: Define sets for numbers divisible by 7 and 3
Let:
- A = set of four-digit numbers divisible by 7
- B = set of four-digit numbers divisible by 3

### Step 4: Calculate the number of elements in set A (divisible by 7)
The first four-digit number divisible by 7 is 1001, and the last is 9996.

To find the number of terms in this arithmetic sequence:
- First term (a) = 1001
- Last term (l) = 9996
- Common difference (d) = 7

Using the formula for the nth term of an arithmetic sequence:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 9996 = 1001 + (n - 1) \cdot 7 \]
\[ 9996 - 1001 = (n - 1) \cdot 7 \]
\[ 8995 = (n - 1) \cdot 7 \]
\[ n - 1 = \frac{8995}{7} \]
\[ n - 1 = 1285 \]
\[ n = 1286 \]

Thus, \( |A| = 1286 \).

### Step 5: Calculate the number of elements in set B (divisible by 3)
The first four-digit number divisible by 3 is 1002, and the last is 9999.

Using the same approach:
- First term (a) = 1002
- Last term (l) = 9999
- Common difference (d) = 3

Using the formula:
\[ 9999 = 1002 + (n - 1) \cdot 3 \]
\[ 9999 - 1002 = (n - 1) \cdot 3 \]
\[ 8997 = (n - 1) \cdot 3 \]
\[ n - 1 = \frac{8997}{3} \]
\[ n - 1 = 2999 \]
\[ n = 3000 \]

Thus, \( |B| = 3000 \).

### Step 6: Calculate the number of elements in the intersection of A and B (divisible by both 7 and 3)
The least common multiple of 7 and 3 is 21.

The first four-digit number divisible by 21 is 1008, and the last is 9996.

Using the same approach:
- First term (a) = 1008
- Last term (l) = 9996
- Common difference (d) = 21

Using the formula:
\[ 9996 = 1008 + (n - 1) \cdot 21 \]
\[ 9996 - 1008 = (n - 1) \cdot 21 \]
\[ 8988 = (n - 1) \cdot 21 \]
\[ n - 1 = \frac{8988}{21} \]
\[ n - 1 = 428 \]
\[ n = 429 \]

Thus, \( |A \cap B| = 429 \).

### Step 7: Apply the principle of inclusion-exclusion to find the number of elements in A union B
Using the formula:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Substituting the values we found:
\[ |A \cup B| = 1286 + 3000 - 429 = 4857 \]

### Step 8: Calculate the number of four-digit numbers that are divisible by neither 7 nor 3
The number of four-digit numbers that are divisible by neither 7 nor 3 is:
\[ \text{Total four-digit numbers} - |A \cup B| \]
\[ = 9000 - 4857 = 4143 \]

### Final Answer
Thus, the number of four-digit numbers that are divisible by neither 7 nor 3 is **4143**.