MAT01 · Number System | Olive Defence

Covers: Classification of Numbers, Divisibility Rules, LCM & HCF, Prime Numbers & Factors, Unit Digit & Cyclicity, Remainders, Fractions & Decimals, Surds & Indices, Number Series & Formulae — all topics tested in defence entrance exams.

Section 1 — Classification of Numbers

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Types of Numbers — Complete Hierarchy

Natural · Whole · Integer · Rational · Irrational · Real

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Type	Definition	Examples
Natural (N)	Counting numbers starting from 1	1, 2, 3, 4 …
Whole (W)	Natural numbers + zero	0, 1, 2, 3 …
Integers (Z)	Whole + negative numbers	… −2, −1, 0, 1, 2 …
Rational (Q)	Expressible as p/q, q ≠ 0	½, −3, 0.75, 0.33…
Irrational	Non-terminating, non-repeating	√2, π, e, √3
Real (R)	All rational + irrational	Any point on number line

Key Subset Relationships

N ⊂ W ⊂ Z ⊂ Q ⊂ R — every natural number is also whole, integer, rational, and real
0 is whole but NOT natural · −5 is integer but NOT whole or natural
Every integer is rational — e.g., −3 = −3/1
π is irrational — 22/7 is only a rational approximation, not equal to π
√4 = 2 is rational; √2 is irrational — perfect squares have rational roots

Even, Odd, Prime, Composite — Critical Definitions

Even: divisible by 2 → 0, 2, 4, 6 … Note: 0 is even
Prime: exactly 2 factors (1 and itself) → 2, 3, 5, 7, 11 … 1 is NOT prime
Composite: more than 2 factors → 4, 6, 8, 9 … 1 is neither prime nor composite
Co-prime: two numbers whose HCF = 1 — need NOT be individually prime (e.g., 8 and 15)
2 is the only even prime number

⚑ Classification Traps

1 is neither prime nor composite — the most common mistake
0 is even, whole but NOT natural
22/7 ≠ π — 22/7 is rational; π is irrational
Every prime > 2 is odd, but not every odd is prime (e.g., 9 = 3 × 3)

Section 2 — Divisibility Rules

Divisibility Rules — 2 through 13

Check divisibility in seconds without long division

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Divisor	Rule	Example
2	Last digit even (0,2,4,6,8)	348 ✓
3	Sum of digits divisible by 3	123: 1+2+3=6 ✓
4	Last 2 digits divisible by 4	1732: 32÷4=8 ✓
5	Last digit is 0 or 5	445, 230 ✓
6	Divisible by BOTH 2 AND 3	126 ✓
8	Last 3 digits divisible by 8	1256: 256÷8=32 ✓
9	Sum of digits divisible by 9	729: 7+2+9=18 ✓
11	(Sum of odd-position digits) − (sum of even-position digits) = 0 or multiple of 11	121: (1+1)−2=0 ✓
12	Divisible by BOTH 3 AND 4	144 ✓

⚑ Divisibility Traps

Div by 6 = div by 2 AND div by 3 — BOTH conditions must hold simultaneously
Div by 4 → check LAST TWO digits (not digit sum)
Div by 8 → check LAST THREE digits
Divisible by 9 → also divisible by 3, but NOT vice versa

Section 3 — LCM & HCF

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HCF & LCM — Methods, Formulae & Applications

Highest Common Factor · Lowest Common Multiple · word problems

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Core Definitions & Key Formula

HCF (GCD): Largest number that divides all given numbers exactly
LCM: Smallest number divisible by all given numbers
Key formula (two numbers only): HCF × LCM = Product of the two numbers
HCF always ≤ smallest number · LCM always ≥ largest number
HCF of fractions = HCF of numerators ÷ LCM of denominators
LCM of fractions = LCM of numerators ÷ HCF of denominators

Exam-Ready Word Problem Patterns

Bells/signals ringing together: answer = LCM of time intervals
Largest square tile for a rectangular floor: answer = HCF of dimensions
Least number divisible by x, y, z leaving remainder r: LCM(x,y,z) + r
Largest number dividing a, b, c with same remainder: HCF of all pairwise differences
Largest number dividing a, b, c leaving remainders p, q, r: HCF(a−p, b−q, c−r)

⚑ LCM/HCF Traps

HCF × LCM = product formula applies to EXACTLY TWO numbers only
HCF of fractions uses LCM of denominators in the denominator — not HCF

Section 4 — Primes & Number of Factors

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Prime Numbers, Factorisation & Counting Divisors

25 primes below 100 · testing primality · factor formula

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All 25 Primes up to 100

1–20: 2, 3, 5, 7, 11, 13, 17, 19
21–50: 23, 29, 31, 37, 41, 43, 47
51–100: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Number of Factors Formula

If N = 2^a × 3^b × 5^c × … then:

Total factors = (a+1)(b+1)(c+1)…
Example: 360 = 2³ × 3² × 5¹ → factors = (3+1)(2+1)(1+1) = 4×3×2 = 24
Perfect square has an odd number of total factors
To test if n is prime: check divisibility by all primes up to √n

Section 5 — Unit Digit & Cyclicity

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Unit Digit of Powers — Cyclicity Method

Find the last digit of any power in under 10 seconds

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Unit digit of base	Cycle (successive powers)	Period
0, 1, 5, 6	Always the same unit digit	1
4	4, 6, 4, 6 …	2
9	9, 1, 9, 1 …	2
2	2, 4, 8, 6, 2, 4, 8, 6 …	4
3	3, 9, 7, 1, 3, 9, 7, 1 …	4
7	7, 9, 3, 1, 7, 9, 3, 1 …	4
8	8, 4, 2, 6, 8, 4, 2, 6 …	4

4-Step Method

Step 1: Identify the unit digit of the base
Step 2: Find the cycle length for that digit
Step 3: Divide the power by the cycle length → find the remainder
Step 4: That remainder's position in the cycle = unit digit. If remainder = 0, use the last value in the cycle
Example — unit digit of 7⁵²: cycle (7,9,3,1), period 4. 52÷4 = rem 0 → last in cycle = 1
Example — unit digit of 2⁸³: cycle (2,4,8,6), period 4. 83÷4 = rem 3 → 3rd in cycle = 8

Section 6 — Remainders

Remainder Rules & Pattern Method

Key properties · pattern/cyclicity · Wilson's theorem

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Fundamental Properties

R[(a+b) ÷ n] = R[ R(a÷n) + R(b÷n) ] ÷ n
R[(a×b) ÷ n] = R[ R(a÷n) × R(b÷n) ] ÷ n
Remainder when 10ⁿ ÷ 9 = always 1
n! ÷ p = 0 (remainder zero) when p is prime and p ≤ n
Wilson's Theorem: If p is prime → (p−1)! ≡ −1 (mod p), i.e., remainder = p−1
aⁿ + bⁿ is divisible by (a+b) when n is odd
aⁿ − bⁿ is divisible by (a−b) for all natural n

Pattern Method — Most Used in Exams

Find remainder of 2¹⁰⁰ ÷ 3: pattern: 2¹→2, 2²→1, 2³→2, cycle of 2. 100 is even → rem = 1
Find unit digit = find remainder ÷ 10 (same operation, different framing)
Remainder when (1!+2!+3!+…+100!) ÷ 10: n! for n≥5 has factors 2 and 5 so ends in 0. Only first four terms matter: 1+2+6+24 = 33 → remainder = 3

Section 7 — Surds & Indices

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Laws of Indices & Surd Operations

All index laws · rationalising · comparing surds

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Laws of Indices — All Rules

aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
(ab)ⁿ = aⁿbⁿ
a⁰ = 1 for any a ≠ 0 · 0⁰ is undefined
a⁻ⁿ = 1/aⁿ · a^(m/n) = ⁿ√(aᵐ)

Surds — Key Operations

Like surds: same radicand → add/subtract: 3√2 + 5√2 = 8√2
Unlike surds: cannot simplify: √2 + √3 ≠ √5
Rationalising: 1/(a−√b) → multiply by (a+√b)/(a+√b); denominator = a²−b
Comparing surds: make indices equal. ∛2 vs ⁴√3 → ¹²√16 vs ¹²√27 → ⁴√3 > ∛2

⚑ Surds & Indices Traps

√(a²+b²) ≠ a+b — extremely common error
√a + √b ≠ √(a+b)
0⁰ is undefined — not 1 and not 0
a⁻¹ = 1/a, NOT −a

Section 8 — Number Series & Key Formulae

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Sum Formulae, AP & GP

All standard summation results tested in defence exams

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Standard Sum Formulae

Sum of first n natural numbers: n(n+1)/2
Sum of first n odd numbers: 1+3+5+…+(2n−1) = n²
Sum of first n even numbers: 2+4+…+2n = n(n+1)
Sum of squares: 1²+2²+…+n² = n(n+1)(2n+1)/6
Sum of cubes: 1³+2³+…+n³ = [n(n+1)/2]²

Arithmetic & Geometric Progressions

AP nth term: aₙ = a + (n−1)d
AP Sum: Sₙ = n/2 × [2a + (n−1)d] = n/2 × (first + last)
GP nth term: aₙ = ar^(n−1)
GP Sum (finite): Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1
GP Sum (infinite, |r|<1): S∞ = a/(1−r)

Scan all high-yield Number System formulae and rules before your exam.

🔢 Number Types

N ⊂ W ⊂ Z ⊂ Q ⊂ R
0 = whole, NOT natural · 1 = neither prime nor composite
2 = only even prime
π is irrational · 22/7 is only an approximation
√4=2 rational · √2 irrational
Co-prime: HCF=1 (need not be individually prime)

÷ Divisibility Rules

2: last digit even · 3: digit sum ÷ 3
4: last 2 digits ÷ 4 · 8: last 3 digits ÷ 8
9: digit sum ÷ 9 · 5: ends in 0 or 5
6 = div by 2 AND 3 · 12 = div by 3 AND 4
11: |odd-pos sum − even-pos sum| = 0 or 11

🔄 LCM & HCF

HCF × LCM = a × b (two numbers only)
HCF ≤ smallest · LCM ≥ largest
HCF(fractions) = HCF(num)/LCM(den)
LCM(fractions) = LCM(num)/HCF(den)
Bells ring together → LCM of intervals
Largest tile → HCF of dimensions
Least no. leaving rem r → LCM + r

⁰ Unit Digit Cycles

0,1,5,6 → always same · 4,9 → period 2
2: (2,4,8,6) · 3: (3,9,7,1) · period 4
7: (7,9,3,1) · 8: (8,4,2,6) · period 4
Divide power by period → find remainder → position in cycle
If remainder = 0 → use last value of cycle

√ Surds & Indices

aᵐ·aⁿ = aᵐ⁺ⁿ · (aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 (a≠0) · 0⁰ = undefined
a^(m/n) = ⁿ√aᵐ · a⁻ⁿ = 1/aⁿ
√a·√b = √(ab) · √a+√b ≠ √(a+b)
Rationalise: × conjugate; denom = a²−b
Compare: make indices equal, compare bases

📐 Sum Formulae

Σn = n(n+1)/2
Σ odd (n terms) = n²
Σ even (n terms) = n(n+1)
Σn² = n(n+1)(2n+1)/6
Σn³ = [n(n+1)/2]²
AP: aₙ = a+(n−1)d · Sₙ = n/2(a+l)
GP∞: S = a/(1−r), |r|<1

% Remainders

R(a+b)÷n = R[R(a)+R(b)]÷n
10ⁿ ÷ 9 → remainder always 1
n! ÷ p (prime ≤ n) → remainder 0
Wilson: (p−1)! mod p = p−1 for prime p
aⁿ+bⁿ div by (a+b) when n is odd
aⁿ−bⁿ always div by (a−b)

🔎 Primes & Factors

Primes to 50: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47
Test: divide by primes up to √n
N=2ᵃ·3ᵇ·5ᶜ → factors = (a+1)(b+1)(c+1)
Perfect square ↔ odd number of total factors
Twin primes: (3,5)(5,7)(11,13)(17,19)(29,31)

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Number System —Types, Properties & Tricks

Number System —
Types, Properties & Tricks