Number of 4 -digit numbers that are less than or equal to 2800 and...

Number of

4

-digit numbers that are less than or equal to

2800

and either divisible by

3

or by

11

, is equal to _____ .

Correct answer

Step-by-step solution

Four-digit numbers that are less than or equal to $2800$ are \left\{1000, 1001, 1002, . . . . , 2800\right\} $Numbers divisible by$ 3 $are \left\{1002, 1005, . . . . , 2796, 2799\right\}$

Let total numbers be $n$ , then

$1002 + \left(n - 1\right)3 = 2799$

$⇒ n = 600$

Numbers divisible by $11$ are \left\{1001, 1012, . . . , 2794\right\} $Let total numbers be$ m $, then$ 2794 = 1001 + \left(m - 1\right)11 $$ ⇒ m = 164 $Numbers divisible by$ 33 $are\left\{1023, 1056, . . . , 2772\right\}$

Let total numbers be $r$ , then

$2772 = 1023 + \left(r - 1\right)33$

$⇒ r = 54$

Number of

4

-digit numbers that are less than or equal to

2800

and either divisible by

3

or by

11

, is equal to

$= 600 + 164 - 54 = 710$

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