Number system
- Which fraction is equal to \(0.2\overline{3}\)? Answer: D. 7/30 Answer: (D) 7/30 Mixed recurring: (all digits − non-recurring part) over 9 for each recurring digit and 0 for each non-recur... 1
- Which of the following is the largest? Answer: D. sqrt[4]5 Answer: (D) sqrt[4]5 Raise all to the 12th power (LCM of 2, 3, 4, 6): √(2) → 2⁶ = 64 , sqrt[3]3 → 3⁴ = 81 , sqrt[4]5 → 5... 1
- What is the remainder when \(3^{100}\) is divided by 7? Answer: A. 4 Answer: (A) 4 3⁶ = 729 leaves remainder 1 on division by 7. 100 = 6 × 16 + 4, so 3¹⁰⁰ leaves the same remainder as 3⁴ = 81 . 81... 1
- What is the remainder when 1! + 2! + 3! + … + 100! is divided by 12? Answer: B. 9 Answer: (B) 9 From 4! onwards every factorial is divisible by 12 (4! = 24). So the remainder comes from 1! + 2! + 3! = 1 + 2 +... 1
- How many factors of 3600 are perfect squares? Answer: A. 12 Answer: (A) 12 3600 = 2⁴ × 3² × 5² . A square factor uses even powers only. Choices: power of 2 from 0, 2, 4 (3 ways), of 3 fr... 1
- What is the remainder when \(5^{90}\) is divided by 124? Answer: C. 1 Answer: (C) 1 Notice 124 = 125 − 1 = 5³ - 1 , so 5³ leaves remainder 1. 5⁹⁰ = (5³)³⁰ leaves remainder 1³⁰ = 1. 1
- What is the remainder when \(2^{31}\) is divided by 5? Answer: D. 3 Answer: (D) 3 Powers of 2 mod 5 repeat in a cycle of 4: 2, 4, 3, 1. 31 = 4 × 7 + 3 , so 2³¹ equiv 2³ = 8 equiv 3 ±od 5 . 1
- What is the largest four-digit number which, when divided by 12, 15 and 18, leaves a remainder of 7 in each case? Answer: C. 9907 Answer: (C) 9907 LCM(12, 15, 18) = 180. The largest 4-digit multiple of 180: 9999 ÷ 180 = 55 remainder 99, so it is 180 × 55... 1