If U = {1, 2, ..., 84}, A = {x ∈ U : 7 divides x}, and B = {x ∈ U : 12 divides x}, what is |A ∪ B|?
Answer and explanation
Correct answer: 18
There are floor(84/7) = 12 multiples of 7 in U, so |A| = 12. There are floor(84/12) = 7 multiples of 12, so |B| = 7. A number belongs to both sets when it is a multiple of lcm(7, 12). Since 7 and 12 are relatively prime, their least common multiple is 84, giving one common element. Therefore, |A ∪ B| = 12 + 7 − 1 = 18.
Frequently asked questions
What is the correct answer to this question?
18
Why is this the correct answer?
There are floor(84/7) = 12 multiples of 7 in U, so |A| = 12. There are floor(84/12) = 7 multiples of 12, so |B| = 7. A number belongs to both sets when it is a multiple of lcm(7, 12). Since 7 and 12 are relatively prime, their least common multiple is 84, giving one common element. Therefore, |A ∪ B| = 12 + 7 − 1 = 18.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
