UPSC NDA I 2016 Mathematics Solved Paper

Concept:Use the inclusion-exclusion principle to count numbers divisible by at least one of 10, 15, or 25, then subtract from total.Explanation:Total numbers from 1 to 1000: $1000$Define sets:$A$: numbers divisible by 10, $B$: by 15, $C$: by 25$n(A) = \frac{1000}{10} = 100$$n(B) = \frac{1000}{15} = 66$ (floor division)$n(C) = \frac{1000}{25} = 40$For intersections, find LCM:LCM(10,15) = 30, so $n(A \cap B) = \frac{1000}{30} = 33$LCM(10,25) = 50, so $n(A \cap C) = \frac{1000}{50} = 20$LCM(15,25) = 75, so $n(B \cap C) = \frac{1000}{75} = 13$LCM(10,15,25) = 150, so $n(A \cap B \cap C) = \frac{1000}{150} = 6$Apply inclusion-exclusion:$n(A \cup B \cup C) = 100 + 66 + 40 - 33 - 20 - 13 + 6 = 146$Numbers divisible by at least one: 146Numbers not divisible by any: $1000 - 146 = 854$Answer:854 (Option B)