Question

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to

Hint:

When calculating the number of numbers divisible by 3, 11, or both, use the inclusion-exclusion principle to avoid double-counting the numbers divisible by both. The formula is \( n(3) + n(11) - n(33) \), where \( n(x) \) represents the count of numbers divisible by \( x \).

Answer 1

Correct Answer: 710

The correct answer is 710.
$1000-2799$
Divisible by 3
$1002+(n-1)3=2799$
$n=600$
Divisible by 11
$1-2799\to \left[\frac{2799}{11}\right]=[254]=254$
$1-999=\left[\frac{999}{11}\right]=90$
$1000-2799=254-90=164$
Divisible by 33
$1-2799\to \left[\frac{2799}{33}\right]=84$
$1-999\to \left[\frac{999}{33}\right]=30$
$1000-2799\to 54$
$\therefore n(3)+n(11)-n(33)$
$600+164-54=710$

Answer 2

Step 1: Find the number of 4-digit numbers divisible by 3. The range is from 1000 to 2799. We use the formula for the number of terms in an arithmetic sequence: \[ 1002 + (n - 1) \times 3 = 2799 \] Solving for \(n\): \[ n = 600 \] So, there are 600 numbers divisible by 3 between 1000 and 2799. 
Step 2: Find the number of 4-digit numbers divisible by 11. We use the floor function to find the total number of multiples of 11: \[ \left\lfloor \frac{2799}{11} \right\rfloor = 254 \] \[ \left\lfloor \frac{999}{11} \right\rfloor = 90 \] Therefore, the number of numbers divisible by 11 between 1000 and 2799 is: \[ 254 - 90 = 164. \] 
Step 3: Find the number of 4-digit numbers divisible by both 3 and 11 (i.e., divisible by 33). We again use the floor function: \[ \left\lfloor \frac{2799}{33} \right\rfloor = 84 \] \[ \left\lfloor \frac{999}{33} \right\rfloor = 30 \] So, the number of numbers divisible by 33 between 1000 and 2799 is: \[ 84 - 30 = 54. \] 
Step 4: Apply the inclusion-exclusion principle to find the total number of 4-digit numbers divisible by 3 or 11. The formula is: \[ n(3) + n(11) - n(33). \] Substituting the values: \[ 600 + 164 - 54 = 710. \] Thus, the total number of 4-digit numbers divisible by 3 or 11 is 710.