Number system
- How many numbers between 100 and 300 are exactly divisible by 7? Answer: D. 28 Answer: (D) 28 Multiples of 7 up to 299: lfloor 299/7 rfloor = 42 . Up to 100: lfloor 100/7 rfloor = 14 . Count = 42 - 14 = 28... 1
- What is the unit digit of \(3^{65} \times 7^{42}\)? Answer: B. 7 Answer: (B) 7 Unit digits of powers of 3 cycle 3, 9, 7, 1; 65 = 4 × 16 + 1 , so 3⁶⁵ ends in 3. Powers of 7 cycle 7, 9, 3, 1; 42... 1
- What is the remainder when \(7^{45}\) is divided by 5? Answer: D. 2 Answer: (D) 2 7 equiv 2 ±od 5 , so we need 2⁴⁵ bmod 5 . Powers of 2 mod 5 repeat in a cycle of 4: 2, 4, 3, 1. 45 = 4 × 11 + 1 ,... 1
- What is the smallest number that must be added to 2375 so that the sum is exactly divisible by 17? Answer: C. 5 Answer: (C) 5 2375 = 17 × 139 + 12 . Remainder 12, so add 17 - 12 = 5 : 2380 = 17 × 140 . 1
- What is the least number that must be added to 1,000 so that the sum is exactly divisible by 23? Answer: D. 12 Answer: (D) 12 1000 ÷ 23 leaves remainder 11 (23 × 43 = 989). Add 23 − 11 = 12 to get 1012 = 23 × 44. 1
- The sum of three consecutive even numbers is 138. What is the largest of the three numbers? Answer: C. 48 Answer: (C) 48 Middle number = 138 ÷ 3 = 46. The numbers are 44, 46, 48, so the largest is 48. 1
- How many numbers from 1 to 500 are divisible by 3 or 7? Answer: D. 214 Answer: (D) 214 Divisible by 3: ⌊500/3⌋ = 166. By 7: ⌊500/7⌋ = 71. By both (21): ⌊500/21⌋ = 23. By 3 or 7 = 166 + 71 − 23 = 2... 1
- Which of the following numbers is divisible by 11? Answer: C. 4832718 Answer: (C) 4832718 Rule: a number is divisible by 11 if the difference between the sums of its alternate digits is 0 or... 1
- What is the unit digit of \(7^{95}\)? Answer: B. 3 Answer: (B) 3 Unit digits of powers of 7 repeat in a cycle of 4: 7, 9, 3, 1. 95 ÷ 4 leaves remainder 3, so the unit digit is th... 1
- How many prime numbers are there from 1 to 50? Answer: C. 15 Answer: (C) 15 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47: 15 primes. Remember: 25 primes up to 100, 15 of them up... 1
- What is the smallest number that must be added to 2500 to make it exactly divisible by 23? Answer: D. 7 Answer: (D) 7 2500 ÷ 23 leaves remainder 16 (23 × 108 = 2484). Number to add = 23 − 16 = 7. Trap: 16 is the number to subtract,... 1
- What is the sum of the first 30 natural numbers? Answer: B. 465 Answer: (B) 465 n(n+1)/2 = 30 × 31/2 = 465 . 1
- If the four-digit number 54x6 is divisible by 9, what is the digit x? Answer: B. 3 Answer: (B) 3 Digit sum = 5 + 4 + x + 6 = 15 + x must be a multiple of 9. So 15 + x = 18 and x = 3. 1
- A number when divided by 56 leaves a remainder of 29. What will be the remainder when the same number is divided by 8? Answer: B. 5 Answer: (B) 5 Shortcut: 8 divides 56, so the remainder is just 29 ÷ 8 → remainder 5. 1
- How many factors (divisors) does 360 have? Answer: B. 24 Answer: (B) 24 360 = 2³ × 3² × 5¹ . Number of factors = (3 + 1)(2 + 1)(1 + 1) = 24. 1
- What is the largest four-digit number exactly divisible by 88? Answer: C. 9944 Answer: (C) 9944 9999 ÷ 88 leaves remainder 55. Largest = 9999 − 55 = 9944. Note: 9856 is divisible by 88 but is not the lar... 1
- What is the remainder when \(17^{23}\) is divided by 16? Answer: A. 1 Answer: (A) 1 17 = 16 + 1, so 17²³ leaves the same remainder as 1²³ = 1. 1
- What is the unit digit of \(2137^{754}\)? Answer: C. 9 Answer: (C) 9 Only the unit digit 7 matters. Cycle of 7: 7, 9, 3, 1. 754 ÷ 4 leaves remainder 2, so the unit digit is 9. 1
- How many zeros are there at the end of 100! (100 factorial)? Answer: C. 24 Answer: (C) 24 Count the factors of 5: ⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24. Trap: 20 forgets the extra 5s in 25, 50, 75 and 100. 1
- What is the sum of all the factors of 72? Answer: A. 195 Answer: (A) 195 72 = 2³ × 3² . Sum of factors = (1 + 2 + 4 + 8)(1 + 3 + 9) = 15 × 13 = 195. 1
- The sum of the squares of three consecutive natural numbers is 2030. What is the middle number? Answer: D. 26 Answer: (D) 26 Let the numbers be n − 1, n, n + 1. Sum of squares = 3n² + 2 = 2030, so n² = 676. n = 26. Check: 25² + 26² + 27... 1
- \(49^{15} - 1\) is exactly divisible by which of the following? Answer: D. 48 Answer: (D) 48 aⁿ - bⁿ is always divisible by a - b . So 49¹⁵ - 1¹⁵ is divisible by 49 − 1 = 48. 50 = 49 + 1 divides it only w... 1
- A number leaves a remainder of 3 when divided by 6. What is the remainder when the square of the number is divided by 6? Answer: A. 3 Answer: (A) 3 Write the number as 6k + 3. Its square leaves the same remainder as 3² = 9, and 9 ÷ 6 leaves 3. 1
- Which fraction is equal to \(0.\overline{36}\)? Answer: A. 4/11 Answer: (A) 4/11 For a pure recurring decimal, write the repeating block over as many 9s: 36/99 = 4/11 . Trap: 9/25 = 0.36 (... 1