Syllabus mapping
Why number systems come first
SSC’s indicative mathematical syllabus explicitly includes computation of whole numbers, decimals and fractions and relationships between numbers. The ideas in this chapter also support percentages, ratio, averages, interest, time and work, algebra and data interpretation.
| Exam family | Typical use | Required depth |
|---|---|---|
| SSC | Divisibility, factors, HCF/LCM, remainders, digit patterns | Fast exact calculation with clear rules |
| Railways / Police | Elementary arithmetic and number relations | Strong foundations and speed |
| Banking | Number properties inside simplification and data problems | Calculation fluency |
| Defence | Arithmetic reasoning and foundational mathematics | Concept plus timed application |
Learning objectives
By the end, you should be able to
- Classify a number into the correct sets.
- Distinguish factors from multiples.
- Test common divisibility rules.
- Find prime factorisations.
- Calculate HCF and LCM.
- Use the HCF–LCM product relation safely.
- Find simple remainders without full division.
- Recognise repeating unit-digit cycles.
Concept 1
The number families
| Set | Meaning | Examples | Boundary to remember |
|---|---|---|---|
| Natural numbers | Counting numbers | 1, 2, 3, … | Some conventions include 0; an exam should state its convention. This course uses 1 as the first natural number. |
| Whole numbers | Natural numbers together with zero | 0, 1, 2, 3, … | No negative values |
| Integers | Whole numbers and their negatives | …, −2, −1, 0, 1, 2, … | Fractions such as 1/2 are not integers |
| Rational numbers | Numbers expressible as p/q where p and q are integers and q ≠ 0 | −3, 2/5, 0.75, 0.333… | Terminating and recurring decimals are rational |
| Irrational numbers | Numbers not expressible as p/q | √2, √3, π | Decimals neither terminate nor repeat |
| Real numbers | All rational and irrational numbers | Every example above | Represented on the number line |
Nesting: natural ⊂ whole ⊂ integers ⊂ rational ⊂ real. Irrational numbers are real but not rational.
Prime and composite numbers
A prime number is a positive integer greater than 1 with exactly two positive factors: 1 and itself. A composite number has more than two positive factors. Therefore, 1 is neither prime nor composite. The only even prime number is 2.
Co-prime numbers
Two integers are co-prime when their HCF is 1. They need not both be prime: 8 and 15 are co-prime.
Concept 2
Divisibility without long division
| Divisor | Test | Example |
|---|---|---|
| 2 | Last digit is even | 4,738 |
| 3 | Digit sum is divisible by 3 | 5,214 → 12 |
| 4 | Last two digits form a multiple of 4 | 7,316 → 16 |
| 5 | Last digit is 0 or 5 | 9,875 |
| 6 | Divisible by both 2 and 3 | 2,574 is even; digit sum 18 |
| 8 | Last three digits form a multiple of 8 | 41,024 → 024 |
| 9 | Digit sum is divisible by 9 | 72,981 → 27 |
| 10 | Last digit is 0 | 8,190 |
| 11 | Difference between alternating-place sums is 0 or a multiple of 11 | 5,082: (5+8)−(0+2)=11 |
Concept 3
Factors, multiples, HCF and LCM
A factor divides a number exactly; a multiple is produced by multiplying it. For 12, the positive factors are 1, 2, 3, 4, 6 and 12, while 12, 24, 36 and 48 are among its multiples.
Prime factorisation
Write every composite integer as a product of primes. For example, 360 = 2³ × 3² × 5.
HCF
Use each common prime at its lowest exponent. For 72 = 2³×3² and 120 = 2³×3×5, HCF = 2³×3 = 24.
LCM
Use every prime present at its highest exponent. LCM = 2³×3²×5 = 360.
Product relation
For two positive integers a and b: HCF(a,b) × LCM(a,b) = a × b. Do not apply this unchanged to three or more numbers.
Counting factors
If n = pᵃqᵇrᶜ, its number of positive factors is (a+1)(b+1)(c+1). Thus 360 = 2³×3²×5¹ has (3+1)(2+1)(1+1) = 24 positive factors.
Concept 4
Remainders and unit-digit cycles
Division can be written as dividend = divisor × quotient + remainder, where 0 ≤ remainder < divisor. In modular notation, 38 ≡ 3 (mod 5).
Useful remainder rules
- Replace numbers by their remainders before adding or multiplying.
- If a ≡ r (mod m) and b ≡ s (mod m), then a+b ≡ r+s and ab ≡ rs (mod m), reduced again modulo m.
- A remainder can never equal or exceed the positive divisor.
Unit digits repeat in cycles
| Base ending | Unit-digit cycle of powers | Cycle length |
|---|---|---|
| 2 | 2, 4, 8, 6 | 4 |
| 3 | 3, 9, 7, 1 | 4 |
| 4 | 4, 6 | 2 |
| 7 | 7, 9, 3, 1 | 4 |
| 8 | 8, 4, 2, 6 | 4 |
| 9 | 9, 1 | 2 |
| 0, 1, 5, 6 | Same digit repeats | 1 |
Worked methods
Examples with reasoning
1. Greatest number dividing two values with the same remainder
Question: Find the greatest number that divides 187 and 233 leaving the same remainder.
Method: A common remainder disappears on subtraction. Compute 233−187=46. The required greatest divisor is 46.
2. Least number divisible by several numbers
Question: Find the least positive number divisible by 12, 18 and 30.
Method: 12=2²×3, 18=2×3², 30=2×3×5. Take highest exponents: 2²×3²×5=180.
3. Unit digit of a large power
Question: Find the unit digit of 7⁸³.
Method: The cycle 7,9,3,1 has length 4. Since 83 leaves remainder 3 on division by 4, select the third digit: 3.
4. Number of factors
Question: How many positive factors does 540 have?
Method: 540=2²×3³×5. Count=(2+1)(3+1)(1+1)=24.
Exam traps
Errors that cost easy marks
Calling 1 prime
Prime numbers have exactly two positive factors. The number 1 has only one.
Mixing factor and multiple
A factor is no larger than a positive number; multiples continue indefinitely.
Using product relation for three numbers
HCF × LCM = product is guaranteed in this simple form for two positive integers, not an arbitrary list.
Cycle remainder zero
If an exponent is divisible by the cycle length, choose the last item in the cycle—not the first.
Invalid remainder
When dividing by 7, a remainder must be one of 0 through 6.
Assuming co-primes are prime
Co-prime describes a pair’s common factor, not the individual numbers.
Original practice
Check understanding after the lesson
These are original syllabus-aligned questions, not claimed SSC previous-year questions.
Rapid revision
The one-minute sheet
- 1 is neither prime nor composite.
- 2 is the only even prime.
- Terminating and recurring decimals are rational.
- HCF takes the lowest common prime powers.
- LCM takes the highest available prime powers.
- For two positive integers: HCF×LCM=product.
- Factor count of pᵃqᵇ is (a+1)(b+1).
- A remainder is at least 0 and smaller than the divisor.
- Digit-sum tests work for 3 and 9.
- Unit digits usually repeat in cycles of 1, 2 or 4.
Source and authorship ledger
Official syllabus boundary
- SSC CGL notice: indicative Mathematical Abilities syllabus
- SSC CHSL notice: indicative Mathematical Abilities syllabus
The explanations, examples, tables and practice questions on this page are original course material written for India 1947 – Till Today. Reviewed 24 September 2026.
Fractions and decimals: equivalence, comparison, operations and recurring forms.