CAT 2003 Slot 2 — QA Question 20
Answer the following question based on the information given below.
Consider three circular parks of equal size with centres at A1, A2 and A3 respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles A1A2A3, B1B2B3 and C1C2C3, as shown. Three sprinters A, B, and C begin running from points A1, B1 and C1 respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.
Consider the sets Tn = {n, n + 1, n + 2, n + 3, n + 4}, where n = 1, 2, 3, … , 96.
How many of these sets contain 6 or any integral multiple thereof (i.e., any one of the numbers 6, 12, 18, ...)?
Answer & solution
80
- B
81
- C
82
- D
83
n = 1, 2, 3, … , 96 and Tn = {n, n + 1, n + 2, n + 3, n + 4}
n could be either 6k or 6k + 1 or 6k + 2 or 6k + 3 or 6k + 4 or 6k + 5.
When n = 6k, then set Tn will definitely contain a multiple of 6 as it contains n
When n = 6k + 5, then set Tn will contain a multiple of 6 as it contains n + 1 = 6k + 6
When n = 6k + 4, then set Tn will contain a multiple of 6 as it contains n + 2 = 6k + 6
When n = 6k + 3, then set Tn will contain a multiple of 6 as it contains n + 3 = 6k + 6
When n = 6k + 2, then set Tn will contain a multiple of 6 as it contains n + 4 = 6k + 6
However, for every n = 6k + 1, the set Tn will not contain any multiple of 6There will be 16 such sets for k = 0 to 15, for which Tn will not contain a multiple of 6
∴ (96 – 16) = 80 sets contain multiples of 6
Hence, option (a).