Question

How many positive integers less than or equal to 1000 are divisible by 10 but not by 13?

Answer 1

Answer:

Lets create the tools required to solve the problem.

Let the multiples of 7 be A7, multiples of 11 be A11 and multiples of 13 be A13.

Then, n(A7) = 142 (You can count from 7, 14, 21 , ...,994)

Then, n(A11) = 90 (You can count from 11, 22, 33, ... 990)

Then, n(A13) = 76 (You can count from 13, 26, 39, ... 988)

Apart from that,  n(A7∩A11)  = Number of positive multiples of 77 less than 1000

= 12 (You can count from 77, 154, 231, ....924)

Similarly  n(A11∩A13)  = 6 (Count from 143, 286, ... 858)

n(A7∩A13)  = 10 (Count from 91, 182, ... 910)

n(A7∩A11∩A13)  = 0 (7*11*13 > 1000)

So, finally, n(Numbers divisible by 7 , 11 or 13 )

=  n(A7∪A11∪A13)  = n(A7) + n(A11) + n(A13) -  n(A7∩A11)  -  n(A11∩A13)  -  n(A7∩A13)  +  n(A7∩A11∩A13)  [By the principle of inclusion-exclusion]

So,  n(A7∪A11∪A13)  = 142 + 90 + 76 - 12 - 6 - 10 + 0

= 280

But we want the complement of the above set. So, required number is 1000 - 280

= 720

Step-by-step explanation: