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Numerical Ability · AFCAT

NA01 — Number System & Simplification

🔢 Numerical Ability – NA01 AFCAT Level ☆ Low Priority

📌 AFCAT Focus: NA01 tests number types (prime, composite, HCF, LCM), BODMAS simplification, and recurring decimals. AFCAT questions are direct — typically 1–2 per paper. Master HCF×LCM = Product, and the BODMAS order. Average students can score full marks here with formula memorisation alone.

1. Types of Numbers

Fig 1.1 — Number System at a Glance

📐 RATIONAL NUMBERS (can write as p/q)

Natural: 1, 2, 3, 4... (no zero)
Whole: 0, 1, 2, 3... (includes zero)
Integers: ...−2, −1, 0, 1, 2... (no fractions)
Terminating decimals: 0.5, 1.25, 0.75
Recurring decimals: 0.333̄, 0.142857̄

⚡ Natural ⊂ Whole ⊂ Integer ⊂ Rational

🔢 IRRATIONAL NUMBERS (cannot write as p/q)

Non-terminating, non-recurring decimals
Examples: √2, √3, π, e
√(perfect square) = Rational — e.g. √4=2 ✓
√(non-perfect square) = Irrational — e.g. √2 ✗
Rational + Irrational = Irrational

π = 3.14159... √2 = 1.41421... e = 2.71828...

Topic ANumber Types — Complete ReferenceDirect AFCAT Facts

Natural

Counting numbers: 1, 2, 3, 4, 5... Does NOT include 0. Also called positive integers.

Whole

Natural numbers + zero: 0, 1, 2, 3, 4... Whole numbers include 0; natural numbers do not.

Integer

All whole numbers + negative numbers: ..., −3, −2, −1, 0, 1, 2, 3... No fractions or decimals.

Rational

Any number expressible as p/q where q ≠ 0. Includes integers, fractions, terminating & recurring decimals. E.g. 3/4, −2, 0.5, 0.333̄

Irrational

Cannot be written as p/q. Non-terminating, non-recurring decimals. E.g. √2, √3, π, e. √(perfect square) is rational.

Prime

Exactly 2 factors: 1 and itself. Smallest prime = 2 (only even prime). Primes up to 50: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47. Note: 1 is NOT prime.

Composite

More than 2 factors. Smallest composite = 4. All even numbers > 2 are composite. 1 is neither prime nor composite.

Co-prime

Two numbers with HCF = 1. Need not be prime individually. E.g. (8, 9), (14, 15), (4, 9). Any two consecutive integers are always co-prime.

Twin Prime

Prime pairs differing by 2. E.g. (3,5), (5,7), (11,13), (17,19), (29,31), (41,43). The pair (2,3) is NOT twin prime (differ by 1).

Even / Odd

Even: divisible by 2 — (0, 2, 4, 6...). Odd: not divisible by 2 — (1, 3, 5...). Key: Odd + Odd = Even. Even × Odd = Even. Odd × Odd = Odd. Zero is even.

Topic BPlace Value vs Face ValueAFCAT Direct

Face Value

The digit itself, regardless of its position. Face value of 7 in 3,7,452 = 7. Face value never changes with position.

Place Value

The value of a digit based on its position in the number. In 3,7,452 — place value of 7 = 7 × 1,000 = 7,000.

Example

Number: 85,634
8 → place value = 80,000, face value = 8
5 → place value = 5,000, face value = 5
6 → place value = 600, face value = 6
3 → place value = 30, face value = 3
4 → place value = 4, face value = 4
Key: Place value = Face value × Position value (ones=1, tens=10, hundreds=100...)

AFCAT Trap

Q: "What is the difference between place value and face value of 6 in 46,821?"
Place value of 6 = 6,000. Face value = 6. Difference = 6,000 − 6 = 5,994.

HCF & LCM

2. HCF & LCM

Fig 2.1 — HCF vs LCM: Key Differences

HCF — Highest Common Factor

Also called GCD or GCF
Take LOWEST powers of common primes
HCF(12,18): 12=2²×3, 18=2×3²
→ HCF = 2×3 = 6
Divides both numbers exactly
HCF ≤ smaller of the two numbers

LCM — Lowest Common Multiple

Smallest number divisible by both
Take HIGHEST powers of all primes
LCM(12,18): 12=2²×3, 18=2×3²
→ LCM = 2²×3² = 36
Is a multiple of both numbers
LCM ≥ larger of the two numbers

⭐ KEY FORMULA: HCF × LCM = A × B | Always verify with this!

🔧 HCF Applications

Largest tile/piece: Find HCF of dimensions (e.g., largest square tile for a 12×18 room = HCF(12,18) = 6 m)
Largest number that divides: HCF of the numbers
Distribute into equal groups: HCF gives max group size
If HCF of a, b = H, then a = Hx, b = Hy where HCF(x,y) = 1

🔧 LCM Applications

Events repeating together: LCM of their periods (e.g., bells ringing every 4 and 6 min: LCM = 12 min)
Smallest number divisible by both: LCM of the numbers
Least time/distance: Real-life scheduling problems
LCM of co-prime numbers = their product

✎ Worked Example — HCF Real-Life: Largest Tile Size

A room is 12 m × 8 m. Find the largest square tile that can cover the floor exactly (no cutting).

The tile side must divide both 12 and 8 exactly.
HCF(12, 8): 12 = 2² × 3, 8 = 2³
HCF = 2² = 4 m
Number of tiles = (12 × 8) / (4 × 4) = 96/16 = 6 tiles

✔ Largest tile side = 4 m | Tiles needed = 6

✎ Worked Example — LCM: Bells Tolling Together

Three bells toll at intervals of 9, 12 and 15 minutes. If they toll together at 8:00 AM, when will they next toll together?

Step 1: Find LCM(9, 12, 15)
9 = 3² 12 = 2² × 3 15 = 3 × 5
LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 minutes = 3 hours
Step 2: 8:00 AM + 3 hours = 11:00 AM

✔ Answer: 11:00 AM

SIMPLIFICATION & BODMAS

3. Simplification & BODMAS

Fig 3.1 — BODMAS Order of Operations

Brackets

( ) [ ] { }

① First

Orders

Powers, Roots

② Second

Division

③ Equal

Multiply

③ Equal

Addition

④ Equal

Subtract

−

④ Equal

D & M are EQUAL priority — go left to right
e.g. 8 ÷ 2 × 4 = 4 × 4 = 16 (not 1)

A & S are EQUAL priority — go left to right
e.g. 10 − 3 + 2 = 7 + 2 = 9 (not 5)

💡 BODMAS Key Rule: D and M have equal priority — go left to right. Same for A and S. Brackets order: ( ) first, then [ ], then { }. Never skip a step in AFCAT MCQs — always work innermost brackets outward.

✎ Worked Example — BODMAS

Simplify: 18 ÷ 3 × {5 + [4 − (2 + 1)]} − 2²

Step 1 (Inner bracket): (2+1) = 3
Step 2 [Square bracket]: [4−3] = 1
Step 3 {Curly bracket}: {5+1} = 6
Step 4 (Orders): 2² = 4
Step 5 (D&M left to right): 18÷3 = 6, then 6×6 = 36
Step 6 (A&S): 36 − 4 = 32

✔ Answer: 32

DECIMAL FRACTIONS & RECURRING DECIMALS

4. Decimal Fractions & Recurring Decimals

Type	Definition	Example	Fraction Form
Terminating Decimal	Ends after finite digits	0.75, 1.25, 0.5	3/4, 5/4, 1/2
Pure Recurring	All digits after decimal repeat	0.333̅ = 0.̲3̲	Numerator = repeating block; Denominator = 9s equal to block length e.g. 0.̲3̲ = 3/9 = 1/3
Mixed Recurring	Some digits fixed, rest repeat	0.1666̅ = 0.1̲6̲	(Number − Non-repeating) ÷ (9s for repeat, 0s for non-repeat) e.g. 0.1̲6̲ = (16−1)/90 = 15/90 = 1/6

Recurring Decimal Formula:
Pure recurring: 0.̲ab̲ = ab / 99 0.̲abc̲ = abc / 999
Mixed recurring: 0.x̲y̲ = (xy − x) / 90 0.xy̲z̲ = (xyz − xy) / 900

APPROXIMATION & ROUNDING

5. Approximation & Rounding Off

🔢 Rounding Rules

Rounding to nearest 10: Look at units digit. If ≥ 5 → round up; if < 5 → round down.
E.g. 47 → 50 | 43 → 40
Rounding to nearest 100: Look at tens digit. If ≥ 5 → round up.
E.g. 867 → 900 | 832 → 800
Rounding to decimal places: Look at the next digit after required place.
E.g. 3.456 to 2 d.p. → 3.46 | 3.453 to 2 d.p. → 3.45
Significant figures: Count from first non-zero digit.
0.00427 to 2 sig. fig. = 0.0043

🔢 Approximation in AFCAT

Approximate first, eliminate options: If 497 × 203 ≈ 500 × 200 = 1,00,000 → exact answer near 1,00,000
Square roots (approximate): √50 ≈ 7.07 | √200 ≈ 14.14 | √10 ≈ 3.16
When to round up: When purchasing material (always buy more, not less)
When to round down: When distributing items evenly (floor the value)
AFCAT uses approximation to test whether students can estimate quickly without a calculator

✎ Worked Example — Approximation

Approximate: 4,998 × 52 ÷ 25.03 (choose nearest from: 9,990 / 10,400 / 10,000 / 9,600)

Round each value: 4,998 ≈ 5,000 | 52 ≈ 50 | 25.03 ≈ 25
5,000 × 50 ÷ 25 = 2,50,000 ÷ 25 = 10,000

✔ Answer ≈ 10,000

📐 Formula Sheet — NA01

HCF & LCM

HCF × LCM = A × B
HCF: take lowest powers of common primes
LCM: take highest powers of all primes
Co-prime: HCF = 1; LCM = A×B

Number Types

Smallest prime: 2 (only even prime)
1 is neither prime nor composite
Twin primes: differ by 2
Co-prime: HCF = 1

BODMAS Order

Brackets → Orders → Div → Mul → Add → Sub
D & M: left to right equally
A & S: left to right equally
Bracket order: ( ) → [ ] → { }

Recurring Decimals

0.̲a̲ = a/9
0.̲ab̲ = ab/99
0.a̲b̲ = (ab−a)/90
0.ab̲c̲ = (abc−ab)/900

Divisibility Rules

By 2: last digit even
By 3: sum of digits ÷ 3
By 4: last 2 digits ÷ 4
By 9: sum of digits ÷ 9
By 11: (sum odd–sum even) ÷ 11

Squares & Cubes (Quick)

√2 = 1.414 √3 = 1.732 √5 = 2.236
1³=1, 2³=8, 3³=27, 4³=64, 5³=125
6³=216, 7³=343, 8³=512, 9³=729, 10³=1000
∛512=8, ∛729=9, ∛1000=10

📝 Topic-Wise PYQs — NA01

Q1. The HCF of 36 and 84 is: AFCAT PYQ

(a) 6(b) 12(c) 18(d) 24

✔ Answer: (b) 12

36 = 2²×3² | 84 = 2²×3×7. HCF = 2²×3 = 12. (Take lowest powers of common primes.)

Q2. LCM of 12, 18 and 24 is: AFCAT PYQ

(a) 36(b) 48(c) 72(d) 144

✔ Answer: (c) 72

12=2²×3 | 18=2×3² | 24=2³×3. LCM = 2³×3² = 8×9 = 72.

Q3. Simplify: 5 + 3 × 4 − 12 ÷ 3 AFCAT PYQ

(a) 13(b) 9(c) 17(d) 28

✔ Answer: (a) 13

BODMAS: 12÷3 = 4; 3×4 = 12; then 5 + 12 − 4 = 13. Common mistake: doing 5+3 first.

Q4. 0.̲3̲6̲ as a fraction is: ⚡ Tricky

(a) 36/99(b) 4/11(c) 9/25(d) 36/100

✔ Answer: (b) 4/11

Pure recurring 0.̲36̲ = 36/99 = 4/11. (Two repeating digits → denominator = 99.)

Q5. Which of the following is NOT a prime number? AFCAT PYQ

(a) 89(b) 97(c) 91(d) 83

✔ Answer: (c) 91

91 = 7 × 13. Not prime. 83, 89, 97 are all prime. Quick check: try dividing by primes up to √91 ≈ 9.5, so check 2,3,5,7 only.

Q6. If HCF of two numbers is 8 and their LCM is 96, and one number is 32, the other is: AFCAT PYQ

(a) 16(b) 24(c) 48(d) 64

✔ Answer: (b) 24

HCF×LCM = A×B → 8×96 = 32×B → B = 768/32 = 24.

🧠 Quick Memory Chart — NA01

🔢 Number Types

Smallest prime: 2
Only even prime: 2
1 = neither prime/composite
Twin primes: differ by 2
Co-prime: HCF = 1
0 is even, whole, not natural

🔢 HCF & LCM

HCF: lowest powers (common)
LCM: highest powers (all)
HCF × LCM = A × B
Co-prime: LCM = A×B
HCF always divides LCM
LCM ≥ HCF always

🔢 BODMAS

B → O → D → M → A → S
D & M: left to right
A & S: left to right
Bracket: ( ) [ ] { }
Square root: Order (O)
Always innermost first

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