In this guide
Numbers run through the whole CTET maths section. Paper I asks about place value, number names and the four operations; Paper II adds factors, primes, divisibility, HCF and LCM, and integers. And in both papers, the pedagogy questions keep coming back to numbers because that is where children's errors are easiest to see: 3005 for three hundred and five, 52 − 27 = 35, 7 × 0 = 7.
This guide covers the content to Paper II level with worked examples, then the errors and teaching moves. For a deeper treatment of place value alone, see place value and number names.
Place value and face value
- Face value is the digit itself.
- Place value is the digit × the value of its place.
In 4,572, the face value of 5 is 5 and its place value is 500.
The face value of 0 and its place value are both 0, wherever it stands. But 0 matters as a placeholder: it is the reason 305 and 35 are different numbers.
| Indian system | International system |
|---|---|
| 1,00,000 = one lakh | 100,000 = one hundred thousand |
| 10,00,000 = ten lakh | 1,000,000 = one million |
| 1,00,00,000 = one crore | 10,000,000 = ten million |
Building numbers
- Largest 4-digit number with different digits: 9,876.
- Smallest 4-digit number with different digits: 1,023 (0 cannot come first).
- There are 900 three-digit numbers (100 to 999).
Kinds of numbers
| Type | Meaning | Examples |
|---|---|---|
| Natural numbers | Counting numbers | 1, 2, 3, … |
| Whole numbers | Natural numbers and 0 | 0, 1, 2, … |
| Integers | Whole numbers and their negatives | …, −2, −1, 0, 1, 2, … |
| Even | Divisible by 2 | 0, 2, 4, 18 |
| Odd | Not divisible by 2 | 1, 7, 23 |
| Prime | Exactly two factors, 1 and itself | 2, 3, 5, 7, 11 |
| Composite | More than two factors | 4, 6, 9, 15 |
| Co-prime | Two numbers whose only common factor is 1 | 8 and 15 |
| Twin primes | Primes that differ by 2 | 3 and 5, 11 and 13, 17 and 19 |
Key facts: 1 is neither prime nor composite. 2 is the only even prime. There are 25 primes between 1 and 100. Co-primes need not be prime themselves: 8 and 15 are both composite.
Divisibility rules
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit even | 3,478 |
| 3 | Sum of digits divisible by 3 | 471: 4 + 7 + 1 = 12 |
| 4 | Last two digits divisible by 4 | 5,316: 16 |
| 5 | Last digit 0 or 5 | 2,345 |
| 6 | Divisible by both 2 and 3 | 1,254 |
| 8 | Last three digits divisible by 8 | 7,120: 120 = 8 × 15 |
| 9 | Sum of digits divisible by 9 | 7,425: 7 + 4 + 2 + 5 = 18 |
| 10 | Last digit 0 | 980 |
| 11 | Difference between sums of alternate digits is 0 or a multiple of 11 | 7,425: (5 + 4) − (2 + 7) = 0 |
Worked example. Is 7,425 divisible by 9 and by 11? Digit sum = 18, so yes for 9 (7,425 ÷ 9 = 825). Alternate sums are 5 + 4 = 9 and 2 + 7 = 9, difference 0, so yes for 11 (7,425 ÷ 11 = 675).
Factors, multiples, HCF and LCM
- The HCF (highest common factor) is the largest number that divides each given number.
- The LCM (lowest common multiple) is the smallest number that each given number divides.
- For two numbers: HCF × LCM = product of the numbers.
Worked example. Find the HCF and LCM of 24 and 36.
- 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3.
- HCF takes the common factors: 2 × 2 × 3 = 12.
- LCM takes each prime the greatest number of times it appears: 2 × 2 × 2 × 3 × 3 = 72.
- Check: 12 × 72 = 864 and 24 × 36 = 864.
Word problem. Three bells ring every 6, 8 and 12 minutes. They ring together at 9:00. When will they next ring together? LCM(6, 8, 12) = 24, so at 9:24.
A useful sign: "largest size that fits exactly" is usually HCF; "when will they meet again" is usually LCM.
Properties of operations
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | 4 + 7 = 7 + 4 | 4 × 7 = 7 × 4 |
| Associative | (2 + 3) + 5 = 2 + (3 + 5) | (2 × 3) × 5 = 2 × (3 × 5) |
| Identity | a + 0 = a | a × 1 = a |
| Distributive (multiplication over addition) | — | a × (b + c) = a × b + a × c |
Subtraction and division are not commutative or associative: 8 − 3 is not 3 − 8.
Zero facts: a × 0 = 0; 0 ÷ a = 0 when a is not 0; division by zero is not defined.
The distributive property is a mental-maths tool: 25 × 102 = 25 × 100 + 25 × 2 = 2,500 + 50 = 2,550.
Roman numerals
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000.
- A symbol repeated adds: III = 3, XX = 20. I, X, C and M may repeat up to three times; V, L and D never repeat.
- A smaller symbol after a larger one adds: VI = 6, LX = 60.
- A smaller symbol before a larger one subtracts: IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900. I is subtracted only from V and X; X only from L and C; C only from D and M.
Examples: XLIX = 40 + 9 = 49. CDXC = 400 + 90 = 490. MMXXVI = 2,026.
Children's errors with numbers
| Error | What the child is thinking | Teaching response |
|---|---|---|
| Writes "three hundred and five" as 3005 | Writes the parts as heard | Build 305 with bundles on a place-value mat; see the 0 as an empty tens place |
| 52 − 27 = 35 | Subtracts the smaller digit from the larger in each column | Regroup a ten into ones with sticks, then record |
| 45 + 38 = 713 | Does not regroup the ten | Exchange ten ones for a ten bundle |
| Lines up 345 + 23 from the left | Treats numbers as strings of digits | Use a place-value chart with columns labelled |
| 7 × 0 = 7 | Confuses with 7 + 0 | "7 groups of nothing" with empty plates |
| "1 is prime" | Only thinks "divisible by 1 and itself" | Ask how many different factors 1 has |
| "Multiplying always makes bigger" | Generalised from whole numbers | Revisit with fractions later: half of 8 |
Teaching numbers well
- Move from concrete (sticks, beads, base-ten blocks, abacus) to pictorial (drawings, place-value charts) to symbolic.
- Let children build and break numbers: 346 = 3 hundreds + 4 tens + 6 ones = 34 tens + 6 ones.
- Use number lines for order, comparison and later for integers.
- Use games: dice for place value, "guess my number" with clues about factors.
- Connect to money: ₹100 notes, ₹10 notes and ₹1 coins model place value directly.
Practice set
- Place value of 7 in 5,738?
- Difference between the place value and face value of 6 in 4,629?
- Value of LXIX?
- How many primes are there between 1 and 30?
- Smallest number to add to 1,000 to make it divisible by 7?
- LCM of 4, 6 and 10?
- Which pair is co-prime: (12, 18), (8, 15), (9, 21), (14, 35)?
- Largest 4-digit number exactly divisible by 12?
- A child reads 403 aloud as "forty-three". What does the child misunderstand?
Answers
- 700.
- 600 − 6 = 594.
- 50 + 10 + 9 = 69.
- 10: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
- 1,000 ÷ 7 = 142 remainder 6, so add 1 (1,001 = 7 × 143).
- 60.
- (8, 15): HCF is 1.
- 9,999 ÷ 12 = 833 remainder 3, so 9,996.
- The role of 0 as a placeholder: the 4 is in the hundreds place.
What to do next
- Write the divisibility rules for 2 to 11 from memory.
- Solve five HCF-LCM word problems and label each as HCF or LCM before solving.
- For each error in the table, describe the concrete activity in one line.
- Move on to fractions and decimals.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Central Board of Secondary Education website .
Get the next CTET guide by email
New guides every week. No spam, unsubscribe any time.