Before we dive into percentages, profit-loss, or time-speed-distance, we need to be rock solid on the basics — what kinds of numbers exist, and how to quickly tell if one number divides another. This stuff comes up everywhere in aptitude tests, and knowing it cold saves us precious seconds on every single problem.
Types of Numbers
Let’s build up from the simplest to the most complex:
- Natural Numbers (N): 1, 2, 3, 4, … (counting numbers, no zero)
- Whole Numbers (W): 0, 1, 2, 3, … (natural numbers + zero)
- Integers (Z): …, -3, -2, -1, 0, 1, 2, 3, … (whole numbers + negatives)
- Rational Numbers (Q): Any number we can write as p/q where p and q are integers and q ≠ 0. Examples: 1/2, -3/4, 7 (which is 7/1), 0.333… (which is 1/3)
- Irrational Numbers: Numbers that CAN’T be written as p/q. They go on forever without repeating. Examples: √2, √3, π, e
- Real Numbers (R): All rational + irrational numbers combined. Basically every number on the number line.
Quick exam tip: Every natural number is a whole number, every whole number is an integer, every integer is rational, and every rational number is real. It’s like nested boxes.
Prime vs Composite Numbers
- Prime: A number greater than 1 that has exactly 2 factors — 1 and itself. Examples: 2, 3, 5, 7, 11, 13…
- Composite: A number greater than 1 that has more than 2 factors. Examples: 4, 6, 8, 9, 10…
- 1 is neither prime nor composite. This is a classic trap in exams.
- 2 is the only even prime number. Another favorite exam question.
Primes up to 50 (memorize these): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
How to check if a number is prime: We only need to check divisibility up to √n. For example, to check if 97 is prime, √97 ≈ 9.8, so we only check primes up to 9: 2, 3, 5, 7. None of them divide 97, so it’s prime.
Divisibility Rules — The Gold Standard
These rules are absolute gold in aptitude tests. Memorize them and we’ll save 30+ seconds per question.
The Rule for 7 — Step by Step
The rule for 7 trips people up, so let’s walk through it. Take the number, remove the last digit, double it, and subtract from the remaining number. Keep going until we get a small enough number.
Example: Is 1029 divisible by 7?
- Remove last digit (9), double it → 18
- Remaining number: 102
- 102 - 18 = 84
- Is 84 divisible by 7? → 84 ÷ 7 = 12. Yes!
The Rule for 11 — Step by Step
Example: Is 918082 divisible by 11?
- Digits at odd positions (1st, 3rd, 5th from right): 2, 0, 1 → sum = 3
- Digits at even positions (2nd, 4th, 6th from right): 8, 8, 9 → sum = 25
- Difference: 25 - 3 = 22, which is divisible by 11. Yes!
Factors and Multiples
- Factor: A number that divides another number evenly. Factors of 12: 1, 2, 3, 4, 6, 12
- Multiple: A number obtained by multiplying. Multiples of 5: 5, 10, 15, 20, 25…
Finding Number of Factors
This is a super common exam question. Here’s the trick:
- Find prime factorization: N = a^p × b^q × c^r
- Number of factors = (p+1)(q+1)(r+1)
Example: How many factors does 360 have?
- 360 = 2³ × 3² × 5¹
- Number of factors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24
Sum of Factors
For N = a^p × b^q:
Sum of factors = [(a^(p+1) - 1)/(a - 1)] × [(b^(q+1) - 1)/(b - 1)]
Example: Sum of factors of 12 (= 2² × 3¹):
- = [(2³ - 1)/(2 - 1)] × [(3² - 1)/(3 - 1)]
- = [7/1] × [8/2]
- = 7 × 4 = 28
- Verify: 1 + 2 + 3 + 4 + 6 + 12 = 28 ✓
Prime Factorization
Breaking a number down into its prime building blocks. We use this everywhere — HCF, LCM, factor counting, you name it.
Method: Keep dividing by the smallest prime that works.
Example: Prime factorize 1260.
- 1260 ÷ 2 = 630
- 630 ÷ 2 = 315
- 315 ÷ 3 = 105
- 105 ÷ 3 = 35
- 35 ÷ 5 = 7
- 7 ÷ 7 = 1
So 1260 = 2² × 3² × 5 × 7
Worked Examples
Example 1: Find the largest 4-digit number divisible by 12.
Largest 4-digit number = 9999. We need it divisible by both 3 and 4.
- 9999 ÷ 12 = 833.25
- 833 × 12 = 9996
- Answer: 9996
Example 2: How many numbers between 1 and 100 are divisible by both 3 and 7?
Divisible by both 3 and 7 means divisible by LCM(3,7) = 21. Multiples of 21 up to 100: 21, 42, 63, 84 → 4 numbers
Example 3: A number when divided by 13 leaves a remainder of 11. What’s the remainder when it’s divided by 7?
The number can be written as 13k + 11. Let’s try values:
- k=0: 11 → 11 ÷ 7 = remainder 4
- k=1: 24 → 24 ÷ 7 = remainder 3
- k=2: 37 → 37 ÷ 7 = remainder 2
Hmm, the remainder depends on k. This is a trick question — we can’t determine a unique answer without more information. Watch out for these in exams!
Example 4: How many factors of 720 are even?
720 = 2⁴ × 3² × 5¹ Total factors = (4+1)(2+1)(1+1) = 30 Odd factors (ignore all 2s, use 3² × 5¹) = (2+1)(1+1) = 6 Even factors = 30 - 6 = 24
Example 5: What is the smallest number that when divided by 5, 6, and 8 leaves remainder 2 in each case?
The number = LCM(5, 6, 8) + 2 = 120 + 2 = 122
Common Exam Patterns
- “Divisible by both X and Y” → Find LCM(X, Y) and work with that
- “How many factors” → Prime factorize, then use (p+1)(q+1)(r+1) formula
- “Leaves remainder r when divided by d” → Number = dk + r for some integer k
- “Largest/smallest n-digit number divisible by X” → Divide the boundary number by X, then adjust
- “Is 1 prime?” → No! Trap question. 1 is neither prime nor composite.
Practice Problems
Q1: How many prime numbers are there between 40 and 60?
Q2: Find the total number of factors of 1800.
Q3: What is the smallest 5-digit number divisible by 11?
Answers:
A1: Primes between 40 and 60: 41, 43, 47, 53, 59 → 5 primes
A2: 1800 = 2³ × 3² × 5² → Factors = (3+1)(2+1)(2+1) = 4 × 3 × 3 = 36
A3: Smallest 5-digit number = 10000. 10000 ÷ 11 = 909.09… → 910 × 11 = 10010