How many numbers form 0 to 999 (0 and 999 inclusive) are indivisible by 7 or 11?

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Solution.

Numbers which divisible by 7:

$a_n=a_1+(n-1)d,$

$a_1=0,d=7,a_n=994,$

from here $n=143.$

Numbers which divisible by 11:

$a_n=a_1+(n-1)d,$

$a_1=0,d=11,a_n=990,$

from here $n=91.$

Numbers which divisible by 7 and 11:

$a_n=a_1+(n-1)d,$

$a_1=0,d=77,a_n=924,$

from here $n=13.$

Numbers which divisible by 7 or 11:

$143+91-13=221.$

Numbers which are not divisible by 7 or 11:

$1000-221=779.$

Answer. 779.