Practice Questions: Number System

Introduction

The Number System is a key topic in bank exams like IBPS PO, SBI Clerk, and RBI Assistant. It covers divisibility rules, remainders, factors, multiples, LCM, HCF, and basic operations. A strong grasp of this topic helps in solving arithmetic problems quickly.

Here are 10 practice questions with solutions to help you prepare.

Practice Questions

1. Find the smallest 4-digit number divisible by 12, 18, and 24.

Solution:

• Find the prime powers of each number:
  • 12 = 2² × 3
  • 18 = 2 × 3²
  • 24 = 2³ × 3
• Take the highest power of each prime: 2³ and 3². So LCM = 2³ × 3² = 8 × 9 = 72.
• The smallest 4-digit number is 1000. Divide 1000 by 72 to find how many multiples of 72 are below 1000: 1000 ÷ 72 ≈ 13.88, so the next whole multiple is 14.
• Required number = 72 × 14 = 1008.
• Ans: 1008

2. What is the remainder when 7⁸⁵ is divided by 6?

Solution:

• Compute 7 modulo 6: 7 ≡ 1 (mod 6).
• Therefore 7⁸⁵ ≡ 1⁸⁵ ≡ 1 (mod 6).
• Ans: 1

3. The sum of two numbers is 45, and their HCF is 5. Find how many such pairs exist.

Solution:

• Let the numbers be 5x and 5y, where x and y are co-prime positive integers.
• Given 5x + 5y = 45 ⇒ x + y = 9.
• List positive integer pairs (x, y) with sum 9 and check co-primeness:
  • (1, 8) - co-prime
  • (2, 7) - co-prime
  • (3, 6) - not co-prime (common factor 3)
  • (4, 5) - co-prime
  • Also the reverses (5,4), (7,2), (8,1) - all co-prime
• Counting ordered pairs gives (1,8), (2,7), (4,5), (5,4), (7,2), (8,1) ⇒ 6 pairs.
• Ans: 6

4. Find the unit digit of 3²⁰² + 7¹²⁵.

Solution:

• Cyclicity of 3 in units place: 3, 9, 7, 1 - cycle length 4.
  • 202 ÷ 4 leaves remainder 2 ⇒ units digit of 3²⁰² is the 2nd in cycle = 9.
• Cyclicity of 7 in units place: 7, 9, 3, 1 - cycle length 4.
  • 125 ÷ 4 leaves remainder 1 ⇒ units digit of 7¹²⁵ is the 1st in cycle = 7.
• Sum of unit digits = 9 + 7 = 16 ⇒ unit digit = 6.
• Ans: 6

5. The LCM of two numbers is 48, and their HCF is 4. If one number is 12, find the other.

Solution:

• Use relation: LCM × HCF = product of the two numbers.
• Let the other number be x. Then 48 × 4 = 12 × x ⇒ x = (48 × 4) ÷ 12 = 192 ÷ 12 = 16.
• Ans: 16

6. Find the greatest number that divides 85, 102, and 136 leaving the same remainder each time.

Solution:

• If a number D leaves the same remainder r when dividing the three numbers, then D divides the differences between the numbers.
• Compute pairwise differences: 102 - 85 = 17, 136 - 102 = 34, 136 - 85 = 51.
• The required D is the HCF of these differences. HCF(17, 34) = 17 (and 17 also divides 51).
• Ans: 17

7. If a number divided by 5 leaves remainder 3, what is the remainder when its square is divided by 5?

Solution:

• Let N ≡ 3 (mod 5). Then N² ≡ 3² ≡ 9 ≡ 4 (mod 5).
• Therefore the remainder is 4.
• Ans: 4

8. The sum of three consecutive odd numbers is 147. Find the largest number.

Solution:

• Let the three consecutive odd numbers be (x - 2), x, (x + 2). Their sum = 3x.
• 3x = 147 ⇒ x = 49. The numbers are 47, 49, 51.
• Largest number = 51.
• Ans: 51

9. What is the smallest number to add to 1056 to make it divisible by 23?

Solution:

• Find remainder when 1056 is divided by 23. 23 × 45 = 1035, so remainder = 1056 - 1035 = 21.
• To reach the next multiple of 23 we need to add 23 - 21 = 2.
• Ans: 2

10. The product of two numbers is 2028, and their HCF is 13. Find the number of such pairs.

Solution:

• Write the numbers as 13x and 13y where x and y are co-prime. Then (13x)(13y) = 2028 ⇒ 169xy = 2028 ⇒ xy = 2028 ÷ 169 = 12.
• Find ordered pairs of positive co-prime integers (x, y) with product 12:
  • Factor pairs of 12: (1,12), (2,6), (3,4), (4,3), (6,2), (12,1).
  • Exclude pairs that are not co-prime: (2,6) and (6,2) share factor 2; the rest are co-prime.
• Valid ordered co-prime pairs: (1,12), (3,4), (4,3), (12,1) ⇒ 4 pairs.
• Ans: 4

Conclusion

These questions cover key concepts like LCM, HCF, remainders, and unit digits-essential for bank exams. Keep practising to improve speed and accuracy!