GATE GA General Aptitude Numerical Ability Number Systems Divisibility Remainders Factors Unit Digits

What Is This?

Numerical Ability: Number Systems – Divisibility, Remainders, Factors, Unit Digits is the study of how numbers interact based on their properties. It involves understanding how to determine if one number divides another, finding remainders, identifying factors, and predicting unit digits in multiplication. This topic appears in exams to test your ability to manipulate and understand numerical relationships quickly and accurately.

Why It Matters

This topic is frequently tested in competitive exams like the GRE, GMAT, SAT, and various job aptitude tests. It typically carries moderate to high marks and tests your logical reasoning, pattern recognition, and arithmetic skills. Mastering this topic can significantly boost your overall score.

Core Concepts

1. Divisibility Rules: Specific rules to check if a number is divisible by another (e.g., a number is divisible by 2 if its last digit is even).
2. Remainders: The leftover part when one number is divided by another.
3. Factors: Numbers that divide evenly into another number.
4. Unit Digits: The rightmost digit of a number, crucial for predicting the last digit of a product.

Prerequisites

1. Basic Arithmetic: You need a solid grasp of addition, subtraction, multiplication, and division.
2. Understanding of Place Value: Knowing how the position of a digit affects its value is crucial.

The Rule-Book (How It Works)

Divisibility Rules

• Divisibility by 2: Last digit is even.
• Divisibility by 3: Sum of digits is divisible by 3.
• Divisibility by 5: Last digit is 0 or 5.
• Divisibility by 9: Sum of digits is divisible by 9.
• Divisibility by 10: Last digit is 0.

Remainders

• Definition: If ( a ) is divided by ( b ), the remainder ( r ) is what's left over.
• Formula: ( a = bq + r ), where ( q ) is the quotient.

Factors

• Definition: A factor is a number that divides another number evenly.
• Finding Factors: List all numbers that divide the given number without a remainder.

Unit Digits

• Multiplication Patterns: Memorize the unit digit results for multiplications from 0 to 9.

Exam / Job / Audit Weighting

• Frequency: High
• Difficulty Rating: Intermediate
• Question Type: Multiple Choice, True/False, Fill in the Blanks

Difficulty Level

Intermediate

Must-Know Rules, Formulas, Standards, or Principles

1. Divisibility Rules: Memorize the rules for 2, 3, 5, 9, and 10.
2. Remainder Formula: ( a = bq + r ).
3. Factor Identification: A number ( n ) has factors from 1 to ( n ) that divide ( n ) evenly.

Worked Examples (Step-by-Step)

Easy

Question: Is 1234 divisible by 2? Step 1: Check the last digit.
Step 2: The last digit is 4, which is even.
Answer: Yes, 1234 is divisible by 2.
Rule Applied: Divisibility by 2.

Medium

Question: Find the remainder when 1234 is divided by 5.
Step 1: Divide 1234 by 5.
Step 2: ( 1234 = 5 \times 246 + 4 ).
Answer: The remainder is 4.
Rule Applied: Remainder formula.

Hard

Question: List all factors of 36.
Step 1: Start with 1 and check each number up to 36.
Step 2: Factors are 1, 2, 3, 4, 6, 9, 12, 18, 36.
Answer: The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
Rule Applied: Factor identification.

Common Exam Traps & Mistakes

1. Mistake: Forgetting to check all digits for divisibility by 3 or 9.
2. Wrong Answer: 123 is divisible by 3 because the last digit is 3.

3. Correct Approach: Sum the digits (1+2+3=6), which is divisible by 3.

4. Mistake: Confusing remainders with quotients.

5. Wrong Answer: The remainder of 15 divided by 4 is 3.

6. Correct Approach: ( 15 = 4 \times 3 + 3 ), so the remainder is 3.

7. Mistake: Not listing all factors.

8. Wrong Answer: The factors of 10 are 1, 2, 5, 10.
9. Correct Approach: Check all numbers from 1 to 10.

Shortcut Strategies & Exam Hacks

• Divisibility by 3 and 9: Sum the digits quickly in your head.
• Remainders: Use mental math to subtract multiples.
• Factors: Start from 1 and check divisibility sequentially.

Question-Type Taxonomy

1. True/False: Is 123 divisible by 9?

2. Favored by: SAT, GRE

3. Multiple Choice: What is the remainder when 123 is divided by 5?

4. Favored by: GMAT, Job Aptitude Tests

5. Fill in the Blanks: List all factors of 24.

6. Favored by: Competitive Exams

Practice Set (MCQs)

1. Question: Is 145 divisible by 5?
2. Options: A) Yes, B) No, C) Cannot be determined, D) Only if the sum of digits is 5
3. Correct Answer: A) Yes
4. Explanation: The last digit is 5.

5. Why the Distractors Are Tempting: B) Looks right if you forget the last digit rule.

6. Question: What is the remainder when 125 is divided by 7?

7. Options: A) 1, B) 2, C) 3, D) 4
8. Correct Answer: D) 4
9. Explanation: ( 125 = 7 \times 17 + 4 ).

10. Why the Distractors Are Tempting: A, B, C) Look right if you miscalculate.

11. Question: List all factors of 18.

12. Options: A) 1, 2, 3, 6, 9, 18, B) 1, 2, 3, 6, 9, C) 1, 2, 3, 6, D) 1, 2, 3, 4, 6, 9, 18
13. Correct Answer: A) 1, 2, 3, 6, 9, 18
14. Explanation: Check all numbers from 1 to 18.

15. Why the Distractors Are Tempting: B, C, D) Look right if you miss some factors.

16. Question: What is the unit digit of ( 7 \times 8 )?

17. Options: A) 6, B) 4, C) 2, D) 0
18. Correct Answer: A) 6
19. Explanation: Memorize unit digit multiplication.

20. Why the Distractors Are Tempting: B, C, D) Look right if you misremember.

21. Question: Is 12345 divisible by 3?

22. Options: A) Yes, B) No, C) Cannot be determined, D) Only if the last digit is 3
23. Correct Answer: A) Yes
24. Explanation: Sum of digits is 1+2+3+4+5=15, which is divisible by 3.
25. Why the Distractors Are Tempting: B, C, D) Look right if you forget the sum rule.

30-Second Cheat Sheet

• Divisibility by 2: Last digit even.
• Divisibility by 3: Sum of digits divisible by 3.
• Divisibility by 5: Last digit 0 or 5.
• Remainder Formula: ( a = bq + r ).
• Factors: Check all numbers from 1 to ( n ).
• Unit Digits: Memorize multiplication patterns.

Learning Path

1. Beginner Foundation: Review basic arithmetic and place value.
2. Core Rules: Memorize divisibility rules, remainder formula, and factor identification.
3. Practice: Solve easy to medium problems.
4. Timed Drills: Practice under exam conditions.
5. Mock Tests: Take full-length practice exams.

Related Topics

1. Prime Numbers: Understanding primes helps in identifying factors.
2. Greatest Common Divisor (GCD): Used to find the largest common factor.
3. Least Common Multiple (LCM): Used to find the smallest common multiple.