Number System Aptitude Questions with Solutions
47 Number System questions with hints, worked solutions and time targets — 16 easy · 27 medium · 4 hard.
- Section: Quantitative Aptitude
- Questions: 47
- By difficulty: 16 easy · 27 medium · 4 hard
- Cost: Free, unlimited on every plan
Divisibility, remainders, HCF and LCM, unit digits, factors.
Number System: the essentials
- Divisibility: for 2, 4 or 8, check the last one, two or three digits; for 3 or 9, the digit sum; for 11, the alternating digit sum, which must be 0 or a multiple of 11.
- HCF × LCM = a × b, for two numbers only. The LCM takes the highest power of each prime, the HCF the lowest power of the common primes.
- N = pᵃ × qᵇ × rᶜ has (a + 1)(b + 1)(c + 1) factors.
- Trailing zeros of n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …
- Same remainder on a, b and c: the greatest divisor is the HCF of their differences. Remainder r on several divisors: the number is LCM × k + r.
- To make N divisible by d, add d − remainder, or subtract the remainder.
Unit digits and remainders
- Unit digits of powers repeat every 4: 2 → 2, 4, 8, 6; 3 → 3, 9, 7, 1; 7 → 7, 9, 3, 1; 8 → 8, 4, 2, 6. A last digit of 4 or 9 alternates; 0, 1, 5 and 6 never change. Use the exponent mod 4, reading 0 as 4.
- Reduce the base modulo the divisor first, then find the cycle of its powers. A base one less than the divisor acts as −1: even powers leave 1, odd powers the divisor minus 1.
Example: 13⁴³ ÷ 5. 13 leaves 3, and 3⁴ = 81 leaves 1; 43 = 4 × 10 + 3, so 13⁴³ leaves the same as 3³ = 27, which is 2.
Questions
- Remainder of a large power Medium
What is the remainder when 7¹⁰⁰ is divided by 5? - Unit digit of a product of powers Medium
What is the unit digit of 3⁴⁷ × 7²³? - HCF, LCM and the other number Easy
The HCF of two numbers is 12 and their LCM is 360. If one of the numbers is 72, what is the other? - A missing digit and divisibility by 9 Easy
For which digit x is the four-digit number 45x3 divisible by 9? - Counting the divisors of 360 Medium
How many positive divisors does 360 have? - Trailing zeros in 100! Medium
How many zeros does 100! end with? - Three consecutive odd numbers Easy
The sum of three consecutive odd numbers is 63. What is the largest of them? - Unit digit of 7⁹⁵ − 3⁵⁸ Medium
What is the unit digit of 7⁹⁵ − 3⁵⁸? - Remainder of 2⁵¹ ÷ 7 Medium
What is the remainder when 2⁵¹ is divided by 7? - Largest four-digit multiple of 88 Easy
What is the largest four-digit number exactly divisible by 88? - HCF of 1.75, 5.6 and 7 Medium
Find the HCF of 1.75, 5.6 and 7. - LCM of 22, 54, 108, 135 and 198 Medium
Find the LCM of 22, 54, 108, 135 and 198. - Trailing zeros in 125! Medium
How many zeros are there at the end of 125!? - Remainder 1 with every divisor, but divisible by 13 Hard
Find the least number which, when divided by 6, 7, 8, 9 and 10, leaves a remainder of 1 in each case, but is exactly divisible by 13. - From a remainder of 47 to a remainder mod 19 Medium
On dividing a number by 342, the remainder is 47. What will the remainder be when the same number is divided by 19? - How many prime factors in all Easy
What is the total number of prime factors in the expression 6¹⁰ × 7¹⁷ × 11²⁷? - Same remainder from three numbers Medium
Find the greatest number that will divide 62, 132 and 237 leaving the same remainder in each case. - Unit digit of 3⁶⁵ × 6⁵⁹ × 7⁷¹ Medium
What is the unit digit of 3⁶⁵ × 6⁵⁹ × 7⁷¹? - Even factors of 1440 Medium
How many even factors does 1440 have? - Least number to add Easy
What is the least number that must be added to 1056 so that the sum is exactly divisible by 23? - Which power is the largest Medium
Which of the following is the largest? 2⁶⁰, 3⁴⁰, 4³⁰, 5²⁰ - Value of a surd expression Hard
If x = 1/(2 − √3), find the value of x³ − 2x² − 7x + 5. - Remainder of a factorial sum Medium
What is the remainder when 1! + 2! + 3! + … + 100! is divided by 24? - Finding the second number Easy
The HCF of two numbers is 11 and their LCM is 693. If one of the numbers is 77, what is the other? - The larger of two numbers Medium
The product of two numbers is 4107 and their HCF is 37. What is the larger number? - How many digits in 2⁴⁰ Medium
How many digits are there in 2⁴⁰? (Take log₁₀2 = 0.3010.) - Remainder of 17²⁰⁰ ÷ 18 Medium
What is the remainder when 17²⁰⁰ is divided by 18? - Total factors of 4200 Medium
How many factors does 4200 have? - Sum of all factors of 36 Medium
What is the sum of all the factors of 36? - Greatest four-digit common multiple Medium
What is the greatest four-digit number divisible by 15, 25, 40 and 75? - A missing digit and divisibility by 3 Easy
If the number 517324 is completely divisible by 3, what is the smallest digit that can replace ? - Primes below 50 Easy
How many prime numbers are there between 1 and 50? - Least number to subtract Easy
What is the least number that must be subtracted from 9999 to make it divisible by 45? - Sum of odd numbers between 100 and 200 Medium
What is the sum of all odd numbers between 100 and 200? - A telescoping sum Medium
Find the value of: 1/(1×2) + 1/(2×3) + 1/(3×4) + … + 1/(9×10) - Squares of consecutive odd integers Medium
The difference between the squares of two consecutive odd integers is always divisible by: - Remainder of 6¹⁷ + 17⁶ ÷ 7 Hard
What is the remainder when 6¹⁷ + 17⁶ is divided by 7? - Counting common multiples in a range Medium
How many numbers between 200 and 600 are divisible by 4, 5 and 6? - A divisibility that always holds Medium
For every natural number n, which is the largest of these that always divides 6n² + 6n? - Two remainders at once Medium
What is the smallest number that leaves a remainder of 3 when divided by 5 and a remainder of 5 when divided by 7? - Unit digit of 4⁵⁴ Easy
What is the unit digit of 4⁵⁴? - Solving a power equation Easy
If 5ˣ = 3125, what is the value of 5⁽ˣ⁻³⁾? - Sum of the first 20 even numbers Easy
What is the sum of the first 20 even numbers? - Making 4321 divisible by 19 Easy
What is the least number that must be added to 4321 to make it exactly divisible by 19? - Divisible by both 9 and 11 Easy
What is the largest three-digit number divisible by both 9 and 11? - Remainder of 3²¹ ÷ 5 Easy
What is the remainder when 3²¹ is divided by 5? - Zeros at the end of a product Hard
How many zeros are there at the end of the product 10 × 20 × 30 × … × 100?
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