Let X = n ∈ N : 1 ≤ n ≤ 50. If
A = n ∈ X: n is a multiple of 2 and
B = n ∈ X: n is a multiple of 7, then the number of elements in the smallest subset of X containing both A and B is ________.

Solution

Maths sir · 3 steps

The smallest set that holds both A and B is just their union, so count it without double-counting.

1. Smallest superset. Any set containing both A and B must contain every element of A ∪ B. And A ∪ B itself contains both, so it is the smallest one.
2. Count each set. Multiples of 2 up to 50: n(A) = 25. Multiples of 7 up to 50 are 7, 14, …, 49, so n(B) = 7. Numbers in both are multiples of 14: 14, 28, 42, so n(A ∩ B) = 3.

   n(A ∪ B) = n(A) + n(B) - n(A ∩ B) = 25 + 7 - 3

3. So. The smallest subset of X containing both A and B has 29 elements.

The common slip: 32. Adding n(A) and n(B) without removing the overlap gives 32, because 14, 28 and 42 get counted twice.

The idea to keep. Smallest set containing A and B means the union. Use n(A ∪ B) = n(A) + n(B) - n(A ∩ B), and find the overlap from multiples of the LCM.

Answer from the NTA final answer key. Solution by PrepWiser.