How many numbers are there between 105 and 505 which are divisible by 2 and 5 both
Answers
Answer:
How many numbers between 0 and 500 are divisible by 3 or 5 or 7?
Let A, B and C be the set of numbers between 0 and 500 that are divisible by 3, 5 and 7 respectively.
n(A) = ⌊500/3⌋ = 166
n(B) = ⌊500/5⌋ = 100
n(C) = ⌊500/7⌋ = 71
Simply adding up all these numbers would not help us form our solution. Notice that numbers which are multiples of LCM of 3 and 5 are counted twice. So are multiples of LCM of 3 and 7, and 5 and 7. Multiples of LCM of 3, 5 and 7 are counted thrice!
We need to make sure that each number is counted exactly once. The cardinality of union of sets A, B and C yields precisely that!
n(A ∩ B) = ⌊500/15⌋ = 33
n(B ∩ C) = ⌊500/35⌋ = 14
n(A ∩ C) = ⌊500/21⌋ = 23
n(A ∩ B ∩ C) = ⌊500/105⌋ = 4
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - ( n(A ∩ B) + n(B ∩ C) + n(A ∩ C) ) + n(A ∩ B ∩ C)
or, n(A ∪ B ∪ C) = 166 + 100 + 71 - (33 + 14 + 23) + 4 = 271
So, there are 271 numbers between 0 and 500 that are divisible by either 3 or, 5 or, 7.
Note that we did not account for the number zero! If you wish to include zero as well, increase the count by one. So, that takes the tally to 272!