Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE MainMathematicsSets and Relations

An auditorium has seats numbered sequentially from 1 to 300 . Seats whose numbers are multiples of 3 are designated as VIP seats. However, a VIP seat is considered 'defective' and removed if its number shares a common factor strictly greater than 1 with 35 . The sum of the numbers of all valid (non-defective) VIP seats is _____.

Correct answer

10110

Step-by-step solution

Total seats are numbered 1 to 300 . The VIP seats are those whose numbers are multiples of 3 . Let V be the set of VIP seat numbers: V = 3, 6, 9, , 300 A VIP seat is defective if its number n satisfies gcd (n, 35) > 1 . Since 35 = 5 7 , a seat is defective if its number is divisible by 5 or 7 . We need to find the sum of valid VIP seats, which are multiples of 3 but not divisible by 5 or 7 . Using the Principle of Inclusion-Exclusion, the required sum is: S_ valid = S₃ - S₁₅ - S₂₁ + S₁₀₅ Sum of multiples of 3 up to

Practice Sets and Relations on Quantrex Academy →

More from Sets and Relations

Let S = 1, 2, 3, , 10 . Consider the set X = R : R is an equivalence relation on the set S such that R has exactly 42 elements . Then the number of elements in X is _____________. 2026Consider the relation R on the set -2,-1,0,1,2 defined by (a, b) R if and only if 1+ab > 0 . Then, among the statements: I. The number of elements in R is 17 II. R is an equivalenc 2026Let R = (x, y) N N : _e(x + y) 2 . Then the minimum number of elements, required to be added in R to make it a transitive relation, is __________. 2026Let A = 1, 4, 7 and B = 2, 3, 8 . Then the number of elements, in the relation R = ((a₁, b₁), (a₂, b₂)) ((A B) (A B)) : a₁ + b₂ divides a₂ + b₁ is _______. 2026Let A = 2, 3, 4, 5, 6 . Let R be a relation on the set A A given by (x, y) R (z, w) if and only if x divides z and y w . Then the number of elements in R is _______. 2026Let R be a relation defined on the set 1,2,3,4 1,2,3,4 by R = ((a, b),(c, d)): 2 a+3 b=3 c+4 d . Then the number of elements in R is 2026Let A = 0,1,2, , 9 . Let R be a relation on A defined by (x, y) R if and only if |x-y| is a multiple of 3. Given below are two statements: Statement I : n( R )=36 . Statement II : 2026Let A = -2,-1,0,1,2,3,4 . Let R be a relation on A defined by x Ry if and only if 2 x+y 2 . Let l be the number of elements in R. Let m and n be the minimum number of elements requ 2026 Full Sets and Relations list All JEE Main PYQs