CTET Mathematics –
Number System Notes & MCQs
Complete 2026 Study Guide
Master every Number System concept tested in CTET — natural numbers, LCM, HCF, prime numbers, fractions, decimals, rational & irrational numbers — with 40 exam-pattern MCQs and free practice tests. Exam: 6 September 2026.
300+ Maths MCQs, full-length Paper 1 & Paper 2 mock tests, instant score & performance analytics. Zero cost, zero credit card needed.
1. Why Number System is the Foundation of CTET Mathematics
Walk into any CTET Mathematics question paper and the very first thing you will notice is how many problems trace back to a solid understanding of the Number System. It is not just one topic — it is the scaffolding on which arithmetic, fractions, algebra, data handling, and measurement are all built. Without mastering this chapter, you cannot reliably solve LCM or HCF problems, you will struggle with fractions, and percentage questions will feel unnecessarily hard.
In CTET Paper 1 (Classes 1–5), the Number System appears in straightforward counting, place value, and basic operations questions. In CTET Paper 2 (Classes 6–8), the same foundation is taken further — rational numbers, real numbers, and the relationship between number types become testable. Across both papers, 3 to 5 direct questions come from Number System topics in every exam, and another 5–8 questions on LCM, HCF, fractions, and decimals rely on this foundation.
2. CTET Mathematics Syllabus 2026 – Full Structure
The CTET Maths section carries 30 marks in both Paper 1 and Paper 2. It is equally split between content knowledge and pedagogy. Number System is explicitly listed in both papers, with the depth and complexity increasing from Paper 1 to Paper 2.
| Paper | Level | Number System Topics | Pedagogy Topics |
|---|---|---|---|
| Paper 1 | Classes 1–5 (up to Class 8 standard) | Natural, Whole, Integers; Place value; LCM & HCF; Fractions & Decimals; Patterns; Basic arithmetic operations | Nature of Maths; Teaching strategies; Error analysis; Remedial teaching; Problem-solving; Evaluation |
| Paper 2 | Classes 6–8 (up to Class 10 standard) | Rational & Irrational numbers; Real numbers; Number line; Surds; Exponents; Properties of numbers; Divisibility | All Paper 1 pedagogy topics plus: Mathematical reasoning; Math anxiety; NCF 2005 implications; Van Hiele model; Constructivist maths teaching |
3. Types of Numbers – Complete Classification for CTET
The Number System is best understood as a family of nested sets, where each type of number builds on the one before it. This classification is directly tested in CTET — questions often ask "which of the following is NOT a rational number?" or "√2 belongs to which number type?" Understanding the hierarchy makes these trivially easy.
Numbers we use for counting, starting from 1. They are positive, non-zero, and have no fractions.
Set: {1, 2, 3, 4, 5, ...} → infiniteSmallest: 1No largest number0 NOT included
All natural numbers plus zero. The only difference from natural numbers is the inclusion of 0.
Set: {0, 1, 2, 3, 4, ...} → infiniteSmallest: 00 IS included
All whole numbers including their negatives. No fractions. "Z" comes from German "Zahlen" (numbers).
Set: {..., -3, -2, -1, 0, 1, 2, 3, ...}No smallest or largest0 is neither + nor –
Any number that can be written as a fraction p/q where p, q are integers and q ≠ 0.
3/4 = 0.75–2/50.333... = 1/3–7 = –7/1CANNOT be written as p/q. Decimal goes on forever with no repeating pattern.
√2 ≈ 1.41421...√3 ≈ 1.73205...π ≈ 3.14159...e ≈ 2.71828...All rational and irrational numbers together. Every point on the number line represents a real number.
R = Rational ∪ IrrationalIncludes all above types
A natural number greater than 1 with exactly two distinct factors: 1 and the number itself.
2, 3, 5, 7, 11, 13...2 is only even prime1 is NOT primeA natural number greater than 1 that has more than two factors — i.e., it can be divided by numbers other than 1 and itself.
4, 6, 8, 9, 10, 12...Smallest composite: 4| Property | Natural (N) | Whole (W) | Integer (Z) | Rational (Q) | Irrational | Real (R) |
|---|---|---|---|---|---|---|
| Includes 0 | ✗ | ✓ | ✓ | ✓ | ✗ | ✓ |
| Includes negatives | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ |
| Includes fractions | ✗ | ✗ | ✗ | ✓ | ✗ | ✓ |
| Decimal terminates/repeats | — | — | — | ✓ | ✗ | Both |
| Expressible as p/q | ✓ | ✓ | ✓ | ✓ | ✗ | Most |
4. Place Value and Face Value – CTET Paper 1 Favourite
Place value questions appear in almost every CTET Paper 1 Mathematics section. They seem simple but candidates who have not practised them carefully do make mistakes under exam pressure. Understand this distinction clearly:
📋 Place Value vs Face Value
Face Value = The digit itself (always) Place Value = Face Value × Position ValueExample: In the number 47,382:
- Digit 4 → Face Value = 4 → Place Value = 4 × 10,000 = 40,000
- Digit 7 → Face Value = 7 → Place Value = 7 × 1,000 = 7,000
- Digit 3 → Face Value = 3 → Place Value = 3 × 100 = 300
- Digit 8 → Face Value = 8 → Place Value = 8 × 10 = 80
- Digit 2 → Face Value = 2 → Place Value = 2 × 1 = 2
💡 CTET twist: "Difference between place value and face value of 6 in 56,823?" → Place value = 6,000; Face value = 6; Difference = 5,994
Indian Number System vs International System
| Position | Indian System | International System |
|---|---|---|
| 1 | Ones | Ones |
| 10 | Tens | Tens |
| 100 | Hundreds | Hundreds |
| 1,000 | Thousands | Thousands |
| 10,000 | Ten Thousands | Ten Thousands |
| 1,00,000 | Lakhs | Hundred Thousands |
| 10,00,000 | Ten Lakhs | Millions |
| 1,00,00,000 | Crores | Ten Millions |
5. Divisibility Rules – All 10 You Must Know for CTET
Divisibility rules are speed tools. Instead of actually dividing, you can check divisibility in seconds using these rules. CTET regularly asks: "Which of the following is divisible by 8?" or "Find the smallest number divisible by both 4 and 6." Learn all ten rules cold.
6. Prime & Composite Numbers – What CTET Actually Tests
Prime number questions in CTET are deceptively simple but candidates frequently lose marks by forgetting key facts. The three most-tested facts are:
- 1 is NEITHER prime NOR composite — this appears in almost every CTET paper
- 2 is the ONLY even prime number — every other even number is composite
- The smallest prime is 2, the smallest composite is 4
- Co-prime numbers: two numbers whose HCF is 1 (e.g. 8 and 9 are co-prime despite neither being prime)
- Twin primes: prime pairs differing by 2 (e.g. 3 & 5, 5 & 7, 11 & 13, 17 & 19, 29 & 31)
- There are 25 prime numbers between 1 and 100
- To test if N is prime: check divisibility by all primes up to √N
✍ Prime Numbers 1–100 (Memorise These)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97Quick check trick: To test if 91 is prime → √91 ≈ 9.5 → test 2, 3, 5, 7 → 91 ÷ 7 = 13. So 91 = 7 × 13, NOT prime!
- Primes 1–10: 2, 3, 5, 7 (only 4 primes)
- Primes 10–50: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 (11 primes)
- Primes 50–100: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (10 primes)
Topic-wise CTET Maths MCQs on Prime Numbers, Divisibility, LCM/HCF, Fractions. Register free and start in 60 seconds.
7. LCM and HCF – Methods, Formulas & CTET Tricks
LCM and HCF questions are consistently present in every CTET Mathematics paper. They appear both as direct calculation questions and as application-based word problems ("Find the smallest number of chocolates that can be equally distributed among 12 and 18 children"). Master both concepts and their relationship — it unlocks an entire class of CTET problems.
Key Definitions
HCF (Highest Common Factor) / GCD: The largest number that divides two or more numbers exactly without leaving a remainder.
LCM (Least Common Multiple): The smallest number that is divisible by each of the given numbers without a remainder.
📋 Critical Formulas (Memorise These)
LCM × HCF = Product of the two numbers HCF of fractions = HCF of numerators ÷ LCM of denominators LCM of fractions = LCM of numerators ÷ HCF of denominatorsImportant: LCM × HCF = Product formula holds ONLY for exactly TWO numbers. For three numbers, use prime factorisation.
- LCM is always ≥ HCF
- LCM is always a multiple of HCF
- If two numbers are co-prime, their HCF = 1 and LCM = product
- The HCF of two consecutive integers is always 1
Method 1 – Prime Factorisation
Write each number as a product of prime factors. Then:
- HCF: Take the lowest power of each common prime factor
- LCM: Take the highest power of all prime factors present
🎯 Worked Example: Find LCM & HCF of 36 and 48
36 = 2² × 3² 48 = 2⁴ × 3¹ HCF = 2² × 3¹ = 4 × 3 = 12 (lowest powers of common factors) LCM = 2⁴ × 3² = 16 × 9 = 144 (highest powers of all factors)Verification: LCM × HCF = 144 × 12 = 1,728 = 36 × 48 = 1,728 ✓
Method 2 – Division Method (for HCF)
Divide the larger number by the smaller. Then divide the previous divisor by the remainder. Repeat until remainder = 0. The last divisor is the HCF. This method is faster for large numbers in exam conditions.
CTET Word Problem Types – LCM & HCF
| Scenario | Use | Why |
|---|---|---|
| Smallest number divisible by both A and B | LCM | LCM = smallest number divisible by both |
| Largest number that divides both A and B exactly | HCF | HCF = largest common divisor |
| Three bells ring together. When will they ring together again? | LCM | They align again at LCM of their intervals |
| Largest tile that fits exactly in a room of given dimensions | HCF | Tile size must divide both dimensions |
| Smallest number that when divided by A, B, C leaves remainder R | LCM + R | LCM(A,B,C) + R = answer |
| Largest number that divides A, B, C leaving same remainder | HCF of (A–B), (B–C), (A–C) | Differences reveal the divisor |
8. Fractions – Types, Operations, and CTET Tricks
Fractions are one of the most frequently tested topics in CTET Paper 1 Mathematics, both as standalone content questions and embedded in word problems. They are also a rich source of pedagogy questions — teachers need to know how children commonly misunderstand fractions and how to address those misconceptions.
Types of Fractions
| Type | Definition | Example | CTET Relevance |
|---|---|---|---|
| Proper | Numerator < Denominator | 3/7, 2/5 | Value is always < 1 |
| Improper | Numerator ≥ Denominator | 7/3, 5/2 | Value is always ≥ 1 |
| Mixed | Whole number + Proper fraction | 2⅓, 4¾ | Convert to improper for operations: 2⅓ = 7/3 |
| Like | Same denominator | 3/7 and 5/7 | Add/subtract directly |
| Unlike | Different denominators | 3/4 and 5/6 | Find LCM of denominators first |
| Equivalent | Same value, different forms | 1/2 = 2/4 = 3/6 | Simplify by dividing by HCF of num & denom |
| Unit | Numerator = 1 | 1/2, 1/7, 1/100 | Larger denominator = smaller fraction |
📝 Fraction Operations – Quick Rules
Addition (like): a/c + b/c = (a+b)/c Addition (unlike): a/b + c/d = (ad + bc) / bd Subtraction: Same as addition but with – Multiplication: a/b × c/d = ac/bd Division: a/b ÷ c/d = a/b × d/c = ad/bc (flip and multiply)Comparing fractions: Cross-multiply. a/b vs c/d → compare ad vs bc. Larger cross-product → larger fraction.
Fraction Pedagogy – What CTET Tests About Teaching Fractions
- Children commonly believe 1/8 > 1/4 because 8 > 4 — use paper folding and visual models to show larger denominator = smaller piece
- Use fraction circles, strips, and number lines as concrete manipulatives before introducing symbolic notation
- Real-life contexts (sharing a pizza, dividing a field) help build intuition before abstract operations
- The constructivist approach asks: let students discover equivalent fractions through folding rather than direct instruction
- Error analysis is key: "A student adds 1/2 + 1/3 and gets 2/5" → the student added numerators and denominators separately → remediation through visual models
9. Decimals – Properties, Operations & CTET Questions
Decimals extend the place value system beyond the ones place. Understanding the decimal number line and the relationship between decimals and fractions is crucial for both Paper 1 content questions and teacher-level error analysis questions.
Decimal Place Values
| Position | Left of decimal | Right of decimal | Value |
|---|---|---|---|
| 3rd from decimal | Hundreds (100) | Thousandths (1/1000) | ×100 or ×0.001 |
| 2nd from decimal | Tens (10) | Hundredths (1/100) | ×10 or ×0.01 |
| 1st from decimal | Ones (1) | Tenths (1/10) | ×1 or ×0.1 |
🎯 Key Decimal Rules for CTET
Multiplying by 10: Move decimal point ONE place RIGHT (35.6 × 10 = 356) Dividing by 10: Move decimal point ONE place LEFT (35.6 ÷ 10 = 3.56) Fraction → Decimal: Divide numerator by denominator Terminating decimal → Fraction: e.g. 0.75 = 75/100 = 3/4Recurring decimals: 0.333... = 1/3; 0.666... = 2/3; 0.142857... = 1/7
- Comparing decimals: align decimal points and compare digit by digit
- 0.5 = 0.50 = 0.500 (trailing zeros after decimal do NOT change value)
- 5 ≠ 5.0 in terms of significant figures but = in value
10. Rational & Irrational Numbers – CTET Paper 2 Focus
This topic appears more heavily in CTET Paper 2 (Classes 6–8), but Paper 1 candidates should also understand the basics. Questions typically ask you to identify which number type an expression belongs to, or to simplify expressions involving surds.
Properties of Rational Numbers
- Rational numbers are dense — between any two rationals, there are infinitely many rationals
- The sum, difference, and product of two rationals is always rational
- Division of rationals is rational UNLESS dividing by zero
- Rational + Irrational = Irrational (e.g. 2 + √3 is irrational)
- Rational × Irrational = Irrational (e.g. 3 × √2 = 3√2, irrational)
- Irrational × Irrational = can be Rational (e.g. √2 × √2 = 2, rational)
Common Irrational Numbers to Remember
11. Mathematics Pedagogy – Number System Teaching for CTET
The pedagogy section of CTET Maths (15 marks) tests whether you know not just the content but how to teach it effectively. Number System pedagogy questions are scenario-based — they describe a classroom situation and ask which strategy a constructivist teacher should use. Here are the key principles:
Nature of Mathematics – Key Pedagogical Principles
- Mathematics is abstract yet concrete-first: Children learn number concepts through physical objects before symbols — use beads, counters, abacus, base-10 blocks
- Constructivist Mathematics: Students discover patterns rather than memorise rules — "Why does multiplying by 10 move the decimal?" should be explored, not just stated
- Mathematics anxiety: Many children fear numbers due to rote-learning-based instruction; solution is contextual, activity-based learning
- Error as opportunity: Errors in number work reveal misconceptions — systematic error analysis is the foundation of remedial teaching
- Gender equity: Mathematics ability is not gender-linked; girls underperform due to social conditioning, not aptitude
- NCF 2005 on Maths: Mathematics should develop children's power of reasoning and problem-solving, not produce rule-following calculators
Common Student Errors in Number System – CTET Pedagogy Questions
| Student Error | Likely Misconception | Remediation Strategy |
|---|---|---|
| Says 0.7 > 0.70 because 7 has no zero | Does not understand trailing zeros after decimal | Use number line; show both points land on same mark |
| Writes 1/8 > 1/4 because 8 > 4 | Treats denominator like whole number | Fraction circles; physically cut paper into 4 and 8 pieces |
| Adds 2/3 + 1/4 = 3/7 | Adds numerators and denominators separately | Visual area models showing the fractions before adding |
| Says 1 is prime | Confuses factor count; sees 1 and itself as "two" factors | Explicitly teach: prime needs exactly TWO DISTINCT factors |
| Rounds 4.45 to 4.4 instead of 4.5 | Rounding rule confusion with 5 | Number line placement; reinforce "5 rounds up" rule with pattern recognition |
Teaching Strategies for Number System
- Concrete → Pictorial → Abstract (CPA): All number concepts should be taught with manipulatives first, then drawings/diagrams, then symbols
- Problem-based learning: Present real-world number problems before teaching the algorithm
- Patterns and discovery: "What is the pattern in multiples of 9? (digit sum always 9)"
- Number talks: Short daily discussions where students share different mental math strategies
- Diagnostic assessment: Before remediating, identify the specific misconception through targeted questioning
- Abacus and base-10 blocks: Most effective manipulatives for place value understanding
🎯 Test Your Number System Knowledge
Practice 300+ CTET Maths MCQs on Number System, Fractions, LCM/HCF, Decimals, and Maths Pedagogy. Free access after a quick one-time registration.
✅ Start Free Practice →12. 40 Important CTET Mathematics MCQs – Number System
These 40 MCQs cover all major Number System subtopics tested in CTET 2018–2024. They match the difficulty and scenario-style of actual CTET questions. Click any question to reveal the answer and explanation.
Number Types & Classification (Q1–Q12)
1 Which of the following is NEITHER prime NOR composite?
2 The only even prime number is ___.
3 √2 belongs to which set of numbers?
4 Which of the following statements is TRUE?
5 The number π (pi) is ___.
6 How many prime numbers are there between 1 and 50?
7 Which of the following is an example of co-prime numbers?
8 Which of the following numbers is divisible by both 3 and 4?
9 What is the place value of 5 in 3,51,274?
10 The difference between the place value and face value of 7 in 57,836 is ___.
11 Which of the following is a terminating decimal?
12 Between which two consecutive integers does √50 lie?
LCM, HCF & Divisibility (Q13–Q27)
13 The HCF of 36 and 84 is ___.
14 The LCM of 12, 18, and 24 is ___.
15 If the LCM of two numbers is 180 and their HCF is 12, and one number is 36, find the other.
16 Three bells ring at intervals of 6, 8, and 10 minutes. They ring together at 9:00 AM. When will they next ring together?
17 The smallest 4-digit number exactly divisible by 18 is ___.
18 A number is divisible by 11 if ___.
19 What is the largest number that divides 625, 875, and 1250 leaving the same remainder in each case?
20 The smallest number that when divided by 4, 6, and 9 leaves remainder 2 in each case is ___.
21 Which digit should replace * in 7*2 to make it divisible by 9?
22 If HCF of two numbers is 8 and their LCM is 96, what is the product of the two numbers?
23 A field is 204 m long and 68 m wide. The largest square tile that can exactly cover it has a side of ___.
24 Which of the following is NOT divisible by 6?
25 The HCF of two co-prime numbers is always ___.
26 91 is ___.
27 The LCM of two numbers is always ___ the HCF.
Fractions, Decimals & Rational Numbers (Q28–Q40)
28 What is 3/4 + 5/6?
29 Which of the following fractions is the largest?
30 A student calculates 1/3 ÷ 1/6 and gets 1/18. What error did the student make?
31 0.375 expressed as a fraction in lowest terms is ___.
32 What is the value of 3.6 × 100 ÷ 0.6?
33 Which of the following is a rational number between 1/3 and 1/2?
34 The sum (2+√3) + (2–√3) is ___.
35 When 0.888... is expressed as p/q, what is p + q?
36 A teacher shows that 1/4 < 1/3 using paper strips. This strategy is called ___.
37 A Class 5 student writes: "0.6 is greater than 0.60 because 0.6 has fewer digits." This error indicates ___.
38 According to NCF 2005, the primary goal of teaching Mathematics in school is to ___.
39 In which of the following situations is a teacher using DIAGNOSTIC assessment?
40 A student consistently makes errors when carrying in multi-digit addition. The MOST appropriate next step for the teacher is ___.
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13. Frequently Asked Questions – CTET Maths Number System
How many questions come from Number System in CTET Maths?
Typically 3–5 questions appear directly from Number System topics (types of numbers, LCM, HCF, divisibility, place value). When combined with related arithmetic (fractions, decimals, percentages), this cluster contributes 8–10 marks out of the 15 content marks in CTET Maths. It is the single most productive topic to master first.
What is the difference between CTET Paper 1 and Paper 2 Maths content?
Paper 1 (Classes 1–5): Number System covers natural numbers, whole numbers, integers, place value, LCM/HCF, fractions, decimals, and basic arithmetic — questions at Class 1–8 level complexity. Paper 2 (Classes 6–8): Extends to rational and irrational numbers, real numbers, exponents, surds, and more complex problem types — questions at Class 6–10 level. The pedagogy section (15 marks) covers the same principles for both.
Is 91 a prime number?
No! 91 is a composite number. This is a classic CTET trick question. 91 = 7 × 13. To verify: √91 ≈ 9.5, so check all primes up to 9 (i.e. 2, 3, 5, 7). Since 7 divides 91 exactly (91÷7=13), it is composite.
What is the quickest way to find LCM in a CTET exam?
The fastest method for 2–3 numbers is Prime Factorisation. Write each number as a product of primes, then take the highest power of each prime. For the special case of two numbers, you can also use: Second number = LCM × HCF ÷ First number — which saves calculation when two of the three values are given.
Is 22/7 equal to π?
No. 22/7 ≈ 3.142857... is only an approximation of π ≈ 3.14159265.... They are NOT equal. π is irrational (non-terminating, non-repeating), while 22/7 is rational. CTET has directly tested this fact — "π is rational because 22/7 is used" is a false statement that students are asked to identify.
Where can I get free CTET Maths mock tests online?
MyTestSeries.in offers free CTET Mathematics mock tests covering Number System, Arithmetic, Geometry, Measurement, Data Handling, and Maths Pedagogy. After a free one-time registration at mytestseries.in/free-registration-form/, you instantly access 300+ MCQs and full-length Paper 1 and Paper 2 mock tests with instant scoring and performance analytics.
What books are best for CTET Maths 2026?
NCERT Mathematics textbooks for Classes 1–8 are non-negotiable. For content and practice, Arihant CTET Mathematics (Paper 1 or Paper 2) is the most popular choice. RS Aggarwal's Quantitative Aptitude helps with speed. For pedagogy, a dedicated CTET Pedagogy guide covering error analysis, NCF 2005, and teaching strategies is essential. See our full comparison: Best Books for CTET 2026 →
14. Related Study Resources on MyTestSeries
📚 CTET Subject-Wise Resources
🧠 CDP & Other CTET Notes
🌎 State TET Resources
🌐 External Official Resources
15. CTET 2026 Mock Tests & Study Products – Free Access
All products below are accessible after your free one-time registration. No payment needed at any stage.
CTET Maths – Number System & Arithmetic Test Series
CTET Paper 1 – Complete Mock Test Series (15 Tests)
CTET Paper 2 – Math & Science / Social Science Series
State TET Combo – UPTET, JHTET, REET, Assam TET
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