# How many numbers in the set {1, 2, . . . , 240} are divisible by...

• Nov 9th 2010, 12:05 PM
ASIWYFA
How many numbers in the set {1, 2, . . . , 240} are divisible by...
Hey guys. Ive been having trouble with this question. Can someone show me how its done ?

How many numbers in the set {1, 2, . . . , 240} are divisible by (a) at least ONE of the
numbers 6, 8, or 9? (b) precisely TWO of the numbers 6, 8, or 9?

• Nov 9th 2010, 12:15 PM
Also sprach Zarathustra
• Nov 9th 2010, 12:24 PM
ASIWYFA
I having trouble understanding that wiki article. How do I use it in my situation ?
• Nov 9th 2010, 12:27 PM
teachermath
this problem is a pretty nice one.

you should investigate what happens with de gcd and lcm first and "think about tiny cases" what if we are interested in {1,2,3,...20}?? make a reasoning there and then think about de whole question.
• Nov 10th 2010, 05:00 AM
topspin1617
Let $A_6,A_8,A_9$ be the sets of integers between 1 and 240 that are divisible by 6, 8, and 9 respectively.

We want to know how many of these integers are divisible by at least one of 6, 8, 9; this is the same as asking how many of these integers are in at least one of the sets I defined above.

Saying something is in at least one of those sets is the same as saying that number is in the union of the sets. So we want to know how many elements there are in the set $A_6\cup A_8\cup A_9$.

Apply the Inclusion-Exclusion principle to find $|A_6\cup A_8\cup A_9|$.