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CDS Mathematics

Number System & Simplification

📐 Arithmetic · Chapter MC01 CDS Elementary Mathematics 🎯 High Priority

The Number System is the single largest chapter in CDS Elementary Mathematics, contributing roughly 18–22 questions per paper. Every other chapter — Percentage, Profit & Loss, Algebra, Geometry — rests on your command of numbers. This chapter tests speed, pattern recognition, and the ability to simplify expressions without a calculator. Master it, and you build the foundation for the entire paper.

📌 What CDS tests in this chapter (based on recent papers):
(1) Unit digit and remainder problems; (2) LCM/HCF — by prime factorisation and division, and their applications; (3) Simplification using BODMAS — multi-step expressions with brackets, fractions and square roots; (4) Divisibility rules applied to choose correct answers; (5) Logarithm laws — product, quotient, power rules; (6) Rational vs irrational classification; (7) Decimal & fraction conversions; (8) Square roots and cube roots of perfect squares/cubes.

Topics at a Glance

① Types of Numbers

Natural, whole, integer, rational, irrational, real

② Divisibility Rules

Tests for 2, 3, 4, 5, 6, 8, 9, 11 — shortcut to eliminate options

③ LCM & HCF

Prime factorisation, division method, word-problem applications

④ Fractions & Decimals

Conversion, operations, recurring decimals

⑤ Square & Cube Roots

Perfect squares, long division method, √ in expressions

⑥ BODMAS & Simplification

Order of operations, bracket clearing, mixed expressions

⑦ Logarithms

log laws, log tables, change of base — tested every paper

⑧ Surds & Rationalisation

√a, conjugates, rationalisating denominators

⑨ Unit Digit & Remainder

Cyclicity of unit digits, remainder shortcuts

1. Types of Numbers

1.1

The Number Hierarchy — Know Where Every Number Lives

Foundation for MCQ classification questions and eliminates careless errors

Every number in CDS belongs to one or more of these sets. Understanding the hierarchy helps you answer "which of the following is irrational?" type questions instantly.

Every number you encounter in CDS belongs somewhere in this hierarchy. Rational numbers include all integers, fractions, and decimals that terminate or repeat.

⚠ Classification Traps in CDS:
• 1 is neither prime nor composite. Frequently tested — "How many prime numbers between 1 and 10?" → Answer: 4 (2,3,5,7). Not 5.
• 2 is the only even prime number.
• 0 is whole but not natural. Natural numbers start at 1.
• √4 = 2 (rational), but √2 is irrational. The root of a perfect square is always rational.
• Every integer is rational (e.g., 5 = 5/1). But not every rational is an integer.

2. Divisibility Rules

2.1

Quick Tests — Avoid Long Division in MCQs

Used to eliminate 2–3 options in under 5 seconds

Divisibility rules let you check divisibility mentally. In CDS MCQs, they are used to quickly identify which answer option is divisible by a given number — without performing actual division.

Divisor	Rule	Quick Example
2	Last digit is even (0, 2, 4, 6, 8)	736 ÷ 2 ✓ (last digit 6)
3	Sum of all digits divisible by 3	4+8+3 = 15, divisible by 3 ✓
4	Last TWO digits form a number divisible by 4	1748 → 48 ÷ 4 = 12 ✓
5	Last digit is 0 or 5	245 ✓, 370 ✓
6	Divisible by BOTH 2 and 3	918 → even & 9+1+8=18 ✓
8	Last THREE digits divisible by 8	3816 → 816 ÷ 8 = 102 ✓
9	Sum of all digits divisible by 9	7+2+9 = 18 ÷ 9 = 2 ✓
11	(Sum of odd-position digits) − (Sum of even-position digits) = 0 or multiple of 11	29876: (2+8+6)−(9+7) = 16−16 = 0 ✓
25	Last TWO digits divisible by 25 (i.e., 00, 25, 50, 75)	1375 → 75 ✓

Worked Example — Divisibility by 11

Test: Is 85,096 divisible by 11?
Odd positions (left to right: 1st, 3rd, 5th): 8 + 0 + 6 = 14
Even positions (2nd, 4th): 5 + 9 = 14
Difference = 14 − 14 = 0 → Yes, divisible by 11 ✓

📌 Divisibility by 12, 15, 18: These compound tests appear in CDS.
• Div by 12 = divisible by both 3 and 4.
• Div by 15 = divisible by both 3 and 5.
• Div by 18 = divisible by both 2 and 9.

📋 TOPIC-WISE PYQ

Divisibility — CDS-Pattern Questions

Q1. Which of the following numbers is divisible by 11?

(a) 5,918 (b) 7,361 (c) 8,294 (d) 9,438

Answer: (a) 5,918
Check (a): Odd positions: 5+1 = 6; Even positions: 9+8 = 17; Diff = 6−17 = −11 → divisible by 11 ✓
Check (b): 7+6 = 13, 3+1 = 4, diff = 9 → not divisible. Others similarly fail.

Q2. The number 3x5 is divisible by 9. The digit x is:

(a) 0 (b) 1 (c) 3 (d) 4

Answer: (b) 1
Sum of digits = 3 + x + 5 = 8 + x. For divisibility by 9, 8 + x must equal 9. So x = 1.

3. LCM & HCF

3.1

Concepts, Methods & the Golden Relationship

Both methods — prime factorisation and division — appear in CDS. Word problems are high-frequency.

HCF (Highest Common Factor)

The largest number that divides all given numbers exactly
Also called GCD (Greatest Common Divisor)
Method 1 — Prime Factorisation: Take product of common prime factors with lowest powers
Method 2 — Division Method: Divide larger by smaller; then divide previous divisor by remainder; repeat until remainder = 0. Last divisor = HCF
HCF ≤ smallest of the given numbers

LCM (Least Common Multiple)

The smallest number divisible by all given numbers
Method 1 — Prime Factorisation: Take product of all prime factors with highest powers
Method 2 — Division Method: Divide all numbers simultaneously by common prime factors; multiply all divisors and remaining numbers
LCM ≥ largest of the given numbers

⚡ The Golden Relationship — Most Tested Formula

For TWO numbers A and B: HCF(A, B) × LCM(A, B) = A × B ⟹ LCM = (A × B) / HCF ⟹ HCF = (A × B) / LCM ⟹ The other number = (HCF × LCM) / given number Note: This relationship holds ONLY for two numbers at a time.

For three numbers, this direct formula does NOT apply. Use prime factorisation.

Worked Example — Finding the Other Number

HCF of two numbers is 12 and LCM is 432. One number is 48. Find the other.
Other number = (HCF × LCM) / given = (12 × 432) / 48 = 5184 / 48 = 108.
Verification: HCF(48, 108) = 12 ✓ LCM(48, 108) = 432 ✓

3.2

Word Problem Applications — Where Marks Are Won

CDS consistently tests these three application types

🔔 Events Repeating Together

Use LCM to find when two periodic events coincide again
Bells ringing every 4, 6, 8 min → meet again after LCM(4,6,8) = 24 min
Soldiers stepping in step on LCM of their step-lengths

📦 Largest Equal Division

Use HCF when distributing items into equal groups
Largest tile to cover a floor exactly = HCF of length and width
Largest equal bundles from two lots = HCF of the two quantities

📐 Smallest Number Tests

Smallest number divisible by a, b, c = LCM(a, b, c)
Smallest number that leaves remainder r when divided by a, b, c = LCM(a,b,c) + r
Smallest number that leaves remainders r₁, r₂... (if a−r₁ = b−r₂ = k) = LCM − k

Worked Example — Remainder Application

Find the least number which when divided by 5, 6, 7, 8 leaves a remainder of 3 in each case.
LCM(5, 6, 7, 8): 5 = 5; 6 = 2×3; 7 = 7; 8 = 2³ → LCM = 2³×3×5×7 = 840
Required number = 840 + 3 = 843.
Check: 843 ÷ 5 = 168 rem 3 ✓ 843 ÷ 6 = 140 rem 3 ✓ 843 ÷ 7 = 120 rem 3 ✓ 843 ÷ 8 = 105 rem 3 ✓

📋 TOPIC-WISE PYQ

LCM & HCF — CDS-Pattern Questions

Q3. The LCM of two numbers is 2310 and their HCF is 30. If one of the numbers is 210, then the other number is:

(a) 320 (b) 330 (c) 396 (d) 420

Answer: (b) 330
Other number = (LCM × HCF) / first = (2310 × 30) / 210 = 69300 / 210 = 330.
Check: HCF(210, 330) = 30 ✓ LCM(210, 330) = 2310 ✓

Q4. Three bells toll at intervals of 9, 12, and 15 minutes respectively. If they toll together at 8:00 AM, at what time will they next toll together?

(a) 10:00 AM (b) 10:30 AM (c) 11:00 AM (d) 11:30 AM

Answer: (c) 11:00 AM
LCM(9, 12, 15): 9 = 3², 12 = 2²×3, 15 = 3×5 → LCM = 2²×3²×5 = 180 minutes = 3 hours.
8:00 AM + 3 hours = 11:00 AM.

4. Simplification & BODMAS

4.1

BODMAS — The Only Order That Gives the Right Answer

Roughly 4–6 direct simplification questions per CDS paper

BODMAS defines the strict order in which operations must be performed. Skipping or reversing this order is the single most common source of wrong answers in simplification questions.

⚡ Bracket Order — Always innermost first

Work through brackets in this order: 1. Vinculum (bar over expression) — treated as grouping 2. ( ) Round brackets — innermost first 3. [ ] Square brackets 4. { } Curly brackets Example: {4 + [3 × (2 + 1)]} = {4 + [3 × 3]} = {4 + 9} = 13

In CDS questions, vinculum (overbar) is common. Evaluate overbar expressions first before any brackets.

Worked Example — Multi-step Simplification

Simplify: 18 − [6 − {4 − (8 − 6 + 3)}]
Step 1 (innermost bracket): 8 − 6 + 3 = 5
Step 2 (curly): 4 − 5 = −1
Step 3 (square): 6 − (−1) = 7
Step 4: 18 − 7 = 11

⚠ BODMAS Traps in CDS:
• Division and Multiplication have equal priority — do whichever comes first from left to right. 12 ÷ 3 × 2 = 4 × 2 = 8, NOT 12 ÷ 6 = 2.
• Same for Addition and Subtraction — left to right.
• Negative inside brackets: −(−3) = +3 when bracket is removed.
• Vinculum overrides everything — always first.

📋 TOPIC-WISE PYQ

Simplification & BODMAS — CDS-Pattern Questions

Q5. Simplify: 5 + 5 × 5 − 5 ÷ 5

(a) 5 (b) 24 (c) 29 (d) 25

Answer: (c) 29
Follow BODMAS: Division first: 5 ÷ 5 = 1. Multiplication: 5 × 5 = 25.
Now: 5 + 25 − 1 = 29.
Trap: Doing operations left to right without BODMAS gives 5 = wrong.

Q6. The value of (0.1)² + (0.01)² + 2(0.1)(0.01) is:

(a) 0.0121 (b) 0.121 (c) 0.1201 (d) 0.1221

Answer: (a) 0.0121
Recognise the identity: (a + b)² = a² + 2ab + b². So the expression = (0.1 + 0.01)² = (0.11)² = 0.0121.

5. Fractions, Decimals & Conversion

5.1

Recurring Decimals, Comparison & Operations

Conversion between fraction ↔ decimal is tested directly and inside simplification

📊 Fraction ↔ Decimal Conversions (Memorise)

1/2 = 0.5 | 1/4 = 0.25 | 3/4 = 0.75
1/3 = 0.333... | 2/3 = 0.666...
1/5 = 0.2 | 2/5 = 0.4 | 3/5 = 0.6 | 4/5 = 0.8
1/6 = 0.1666... | 5/6 = 0.8333...
1/7 = 0.142857... | 1/8 = 0.125 | 3/8 = 0.375
1/9 = 0.111... | 1/11 = 0.0909...

🔄 Recurring Decimal → Fraction Rule

Pure recurring: 0.ā = a/9 0.āb̄ = ab/99
Mixed recurring: 0.ab̄ = (ab − a)/90
Example: 0.3̄ = 3/9 = 1/3
Example: 0.1̄8̄ = 18/99 = 2/11
Example: 0.16̄ = (16−1)/90 = 15/90 = 1/6
Denominator: as many 9s as recurring digits, then 0s for non-recurring

⚡ Comparing Fractions — Fastest Method

To compare a/b and c/d: Cross-multiply: compare a×d vs b×c If a×d > b×c → a/b > c/d If a×d < b×c → a/b < c/d Example: Compare 3/7 and 5/11 3 × 11 = 33 vs 7 × 5 = 35 33 < 35 → 3/7 < 5/11

This avoids converting to decimals and is much faster in MCQ settings.

6. Square Roots & Cube Roots

6.1

Perfect Squares, Cubes & Approximation

Needed inside simplification, surd, and geometry questions

Perfect Squares (1² to 20²)

1, 4, 9, 16, 25, 36, 49, 64, 81, 100
121, 144, 169, 196, 225, 256, 289, 324, 361, 400
√144 = 12 | √196 = 14 | √225 = 15
√256 = 16 | √289 = 17 | √361 = 19

Perfect Cubes (1³ to 10³)

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
∛125 = 5 | ∛216 = 6 | ∛343 = 7
∛512 = 8 | ∛729 = 9 | ∛1000 = 10
Key: unit digit of cube matches unit digit of its root for 1,5,6; others have pairs (2↔8, 3↔7, 4↔6)

📌 Key Surd Values to Memorise for CDS:
√2 ≈ 1.414 | √3 ≈ 1.732 | √5 ≈ 2.236 | √6 ≈ 2.449 | √7 ≈ 2.646 | √10 ≈ 3.162
These values appear directly in "approximate value" MCQs and in trigonometry ratios.

7. Logarithms

7.1

Definition, Laws & Applications — Tested Every CDS Paper

2–4 dedicated logarithm questions in almost every CDS paper

⚡ Core Definition & Laws of Logarithms

DEFINITION: logₐ(m) = x ⟺ aˣ = m (read: "log m to base a") a > 0, a ≠ 1, m > 0 KEY LAWS (base a throughout): Law 1 — Product: log(mn) = log m + log n Law 2 — Quotient: log(m/n) = log m − log n Law 3 — Power: log(mⁿ) = n × log m Law 4 — Base swap: log_b(m) = log m / log b (Change of base) SPECIAL VALUES: log_a(a) = 1 (log of base to itself = 1) log_a(1) = 0 (log of 1 to any base = 0) log_a(aⁿ) = n a^(log_a m) = m (anti-log relationship) COMMON LOG (base 10): log 2 ≈ 0.3010 log 3 ≈ 0.4771 log 5 = log(10/2) = 1 − log 2 ≈ 0.6990 log 7 ≈ 0.8451

In CDS, log always means log₁₀ (common logarithm) unless a different base is specified.

Fig 3. Logarithm is simply another way of writing an exponential relationship. The three quantities (base, exponent, result) are always the same in both forms.

Worked Example — Log Laws in Action

If log 2 = 0.3010 and log 3 = 0.4771, find log 12.
log 12 = log(4 × 3) = log 4 + log 3 = log 2² + log 3 = 2 log 2 + log 3
= 2(0.3010) + 0.4771 = 0.6020 + 0.4771 = 1.0791

Find log₄ 64.
64 = 4³, so log₄ 64 = log₄ 4³ = 3 × log₄ 4 = 3 × 1 = 3.

📋 TOPIC-WISE PYQ

Logarithms — CDS-Pattern Questions

Q7. If log₂ (log₂ (log₂ x)) = 2, then x equals:

(a) 256 (b) 512 (c) 65536 (d) 4096

Answer: (c) 65536
log₂(log₂(log₂ x)) = 2 → log₂(log₂ x) = 2² = 4 → log₂ x = 2⁴ = 16 → x = 2¹⁶ = 65536.

Q8. The value of log₁₀ (0.001) is:

(a) 3 (b) −3 (c) 1/3 (d) −1/3

Answer: (b) −3
0.001 = 10⁻³, so log₁₀(10⁻³) = −3. (log_a(aⁿ) = n)

Q9. If log 2 = 0.30103, the number of digits in 2²⁵⁶ is:

(a) 77 (b) 78 (c) 79 (d) 80

Answer: (b) 78
log(2²⁵⁶) = 256 × log 2 = 256 × 0.30103 = 77.06368.
Number of digits = ⌊77.06368⌋ + 1 = 77 + 1 = 78.
Rule: Number of digits in N = ⌊log N⌋ + 1 (floor of log + 1).

8. Surds & Rationalisation

8.1

Simplifying Expressions with Square Roots

Appears in simplification, mensuration and trigonometry sub-questions

A surd is an irrational root that cannot be simplified to a whole number (e.g., √2, √3, ∛5). Rationalisation means removing the surd from the denominator by multiplying by its conjugate.

⚡ Rationalisation — Conjugate Method

Denominator (a + √b) → multiply by (a − √b) Result: (a + √b)(a − √b) = a² − b (rational!) Denominator (√a + √b) → multiply by (√a − √b) Result: a − b (rational!) Example: 1/(√5 − √3) = (√5 + √3) / [(√5 − √3)(√5 + √3)] = (√5 + √3) / (5 − 3) = (√5 + √3) / 2 Example: 1/(2 + √3) = (2 − √3) / [(2)² − (√3)²] = (2 − √3) / (4 − 3) = (2 − √3) / 1 = 2 − √3

📌 Operations on Surds:
• Only like surds can be added/subtracted: 3√2 + 5√2 = 8√2; but 3√2 + 5√3 cannot be combined.
• Multiplication: √a × √b = √(ab) | (√a)² = a
• Simplify first: √48 = √(16×3) = 4√3 before operating.

9. Unit Digit & Cyclicity

9.1

Finding Unit Digits Without Full Calculation

Saves 2–3 minutes per question in competitive exams

The unit digit of powers follows a repeating cycle. You never need to calculate the full number — just find where the exponent falls in the cycle.

Worked Example — Unit Digit

Find the unit digit of 7⁹⁵.
Cycle of 7: 7, 9, 3, 1 (length 4). Divide exponent: 95 ÷ 4 = 23 remainder 3.
3rd position in cycle: 3. Unit digit of 7⁹⁵ = 3.

Find unit digit of 2¹⁰⁰ + 3⁸¹.
2¹⁰⁰: 100 ÷ 4 = 25 rem 0 → position 4 → unit digit = 6.
3⁸¹: 81 ÷ 4 = 20 rem 1 → position 1 → unit digit = 3.
Sum unit digit: 6 + 3 = 9. Answer: unit digit = 9.

🔥 TRICKY QUESTIONS

Number System — Classic CDS Traps

🧩 T1. The number (3²⁵ + 3²⁶ + 3²⁷ + 3²⁸) is divisible by:

Solution: 40.
Factor out 3²⁵: 3²⁵(1 + 3 + 3² + 3³) = 3²⁵(1 + 3 + 9 + 27) = 3²⁵ × 40.
This is divisible by 40. (And also by any factor of 3²⁵ × 40.)
Trap: Trying to compute each term separately — completely unnecessary. Always factor first.

🧩 T2. If log(x + y) = log x + log y, what is the relation between x and y?

Solution: xy = x + y, i.e., 1/x + 1/y = 1.
log(x + y) = log x + log y = log(xy) ⟹ x + y = xy ⟹ xy − x − y = 0
Add 1 to both sides: xy − x − y + 1 = 1 ⟹ (x−1)(y−1) = 1.
Trap: Thinking log(x+y) = log x + log y is always true — it is not. It's only true when xy = x + y.

🧩 T3. The HCF of two numbers is 11 and their LCM is 7700. If one number is 275, find the other.

Solution: 308.
Other = (HCF × LCM) / given = (11 × 7700) / 275 = 84700 / 275 = 308.
Verify: HCF(275, 308) = 11 ✓ LCM(275, 308) = 275×308/11 = 7700 ✓
Trap: Using LCM/HCF directly without applying the product formula.

📐 Formula Sheet — MC01 Number System & Simplification

📊 LCM & HCF

HCF × LCM = A × B (for two numbers)
LCM ≥ any of the numbers; HCF ≤ any
HCF always divides LCM
Coprimes: HCF = 1; LCM = A × B
Smallest no. leaving rem r when div by a,b,c: LCM(a,b,c) + r

📋 Logarithm Laws

log(mn) = log m + log n
log(m/n) = log m − log n
log(mⁿ) = n log m
log_a(a) = 1; log_a(1) = 0
Digits in N = ⌊log₁₀ N⌋ + 1

🔄 Recurring Decimals

0.ā = a/9
0.āb̄ = ab/99
0.ab̄ = (ab − a) / 90
0.abc̄ = (abc − ab) / 900
Rationalise: multiply by conjugate

🔢 Unit Digit Cycles

2: cycle 2,4,8,6 (length 4)
3: cycle 3,9,7,1 (length 4)
7: cycle 7,9,3,1 (length 4)
8: cycle 8,4,2,6 (length 4)
0,1,5,6: always same unit digit

📏 Divisibility Rules

3 & 9: digit sum divisible
4: last 2 digits divisible by 4
8: last 3 digits divisible by 8
11: alternating digit-sum diff = 0 or 11
12 = div by 3 and 4; 15 = div by 3 and 5

📐 Surds

√a × √b = √(ab)
√a / √b = √(a/b)
a√x ± b√x = (a ± b)√x
Rationalise (a+√b): multiply by (a−√b)
√2≈1.414, √3≈1.732, √5≈2.236

⚡ Quick Revision Booster — MC01

🔢 Number Types

1 = neither prime nor composite
2 = only even prime
0 = whole, not natural
√(perfect square) = rational
Integers ⊂ Rational ⊂ Real
π, e, √2, √3 = irrational

🏆 HCF & LCM Shortcuts

HCF × LCM = A × B
Coprimes: LCM = product
Bells/events: use LCM
Tile/equal groups: use HCF
For 3+ numbers: only prime factorisation

📊 Log Values (CDS)

log 2 ≈ 0.3010
log 3 ≈ 0.4771
log 5 ≈ 0.6990 (= 1 − log 2)
log 7 ≈ 0.8451
log 1 = 0; log 10 = 1
log (negative) = undefined

⚡ BODMAS Order

Vinculum → ( ) → [ ] → { }
Then: Powers/Roots
Then: × and ÷ (left to right)
Then: + and − (left to right)
−(−x) = +x when brackets removed

🌀 Unit Digit Strategy

Divide exponent by cycle length
Find remainder → position in cycle
Rem 0 = last value in cycle
5ⁿ always ends in 5; 6ⁿ in 6
For product: find each unit digit, multiply

🚨 Critical CDS Traps

12 ÷ 3 × 2 = 8, NOT 2 (left-to-right)
LCM≠HCF relation: only 2 numbers
log(x+y) ≠ log x + log y generally
√(a²+b²) ≠ a+b (very common error)
0.9̄ = 1 exactly (not less than 1)

✏️ PRACTICE EXERCISE

Attempt These Yourself — Answers on Next Attempt

E1. Find the greatest number that exactly divides 513, 783, and 1107.

(a) 27 (b) 54 (c) 81 (d) 108

💡 Hint: The greatest number that exactly divides all three = HCF(513, 783, 1107). Find by prime factorisation or division method.

E2. Simplify: 0.7̄ − 0.4̄ + 0.3̄

(a) 2/3 (b) 7/9 (c) 5/9 (d) 2/9

💡 Hint: Convert each recurring decimal to fraction first: 0.7̄ = 7/9, 0.4̄ = 4/9, 0.3̄ = 3/9. Then compute.

E3. If log₃ x = 4, then x is:

(a) 12 (b) 64 (c) 81 (d) 34

💡 Hint: Convert to exponential form. log₃ x = 4 means 3⁴ = x.

E4. What is the unit digit of 13⁸¹ + 17⁶³?

(a) 0 (b) 4 (c) 6 (d) 8

💡 Hint: Unit digit depends only on the unit digit of the base. 13 → cycle of 3 (3,9,7,1). 17 → cycle of 7 (7,9,3,1). Find each, then add unit digits.

E5. The HCF of two numbers is 16 and their product is 4096. Find their LCM.

(a) 128 (b) 256 (c) 512 (d) 64

💡 Hint: HCF × LCM = product of the two numbers = 4096. LCM = 4096 / HCF = 4096 / 16.

E6. Simplify: [2(1/4 + 1/5) − 3/5] ÷ 1/5

(a) 1 (b) 3 (c) 5 (d) 7

💡 Hint: Work strictly inside the brackets first. 1/4 + 1/5 = 9/20. Then multiply by 2 = 9/10. Then subtract 3/5 = 9/10 − 6/10 = 3/10. Finally divide by 1/5 = multiply by 5.

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