Numbers and operations
Place value in the Indian and international systems, the divisibility tests and factor rules the paper leans on, integers, fractions and decimals — with the order of operations and the traps that turn a correct method into a wrong answer.
What this heading covers
The number system is the largest content block in either mathematics paper, and it is examined at the level of a well-taught upper primary class rather than a competitive arithmetic test. What decides marks is precision on definitions and the small rules — divisibility, HCF and LCM, the order of operations — not speed.
Place value
The Indian system groups digits as ones, tens, hundreds, then thousands and ten thousands, then lakhs and ten lakhs, then crores.
The international system groups in threes: ones, thousands, millions, billions.
| Indian | International |
|---|---|
| 1 lakh = 1,00,000 | 100 thousand = 100,000 |
| 10 lakh = 10,00,000 | 1 million |
| 1 crore = 1,00,00,000 | 10 million |
| 100 crore | 1 billion |
So 1 million = 10 lakh and 1 crore = 10 million. The commas fall in different places, which is why a question that writes a number in words and asks for the numeral is really testing grouping.
Two distinctions the paper sets:
- Face value is the digit itself. Place value is the digit multiplied by its position. In 5,274 the face value of 2 is 2; its place value is 200.
- The successor of a number is one more, the predecessor one less.
Rounding: look at the digit to the right of the place you are rounding to; 5 or more rounds up.
Roman numerals: I, V, X, L, C, D, M for 1, 5, 10, 50, 100, 500, 1000. A smaller value before a larger one is subtracted (IX = 9, XL = 40); I, X, C can be repeated up to three times; V, L, D are never repeated and never subtracted.
The families of numbers
| Set | Members |
|---|---|
| Natural numbers | 1, 2, 3, … |
| Whole numbers | 0, 1, 2, 3, … |
| Integers | …, −2, −1, 0, 1, 2, … |
So the one difference between natural and whole numbers is zero, and that is asked directly. Every natural number is a whole number; the reverse is not true.
Properties worth stating, because questions name them:
- Closure: whole numbers are closed under addition and multiplication, not under subtraction or division (3 − 5 and 3 ÷ 5 leave the set).
- Commutative: a + b = b + a and a × b = b × a. Subtraction and division are not commutative.
- Associative: grouping does not matter for addition and multiplication.
- Distributive: a × (b + c) = a × b + a × c.
- Identity: 0 for addition, 1 for multiplication.
- Division by zero is not defined. Zero divided by any non-zero number is 0.
Factors, multiples and the rules that save time
A factor divides a number exactly; a multiple is what you get by multiplying it. Every number is a factor and a multiple of itself; 1 is a factor of everything.
Prime numbers have exactly two factors, 1 and themselves. Composite numbers have more. 1 is neither prime nor composite — it has only one factor — and 2 is the only even prime. Both are standard question material.
Co-prime numbers have no common factor except 1 (8 and 15), and they need not themselves be prime.
Divisibility tests
| Divisible by | Test |
|---|---|
| 2 | Last digit is even |
| 3 | Digit sum divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Ends in 0 or 5 |
| 6 | Divisible by 2 and 3 |
| 8 | Last three digits divisible by 8 |
| 9 | Digit sum divisible by 9 |
| 10 | Ends in 0 |
| 11 | Difference of alternating digit sums is 0 or divisible by 11 |
The 3 and 9 tests are the ones papers use most, usually as a starred-digit question: 7*82 is divisible by 9, find * — add the known digits, and choose the digit that lifts the total to the next multiple of 9.
HCF and LCM
The HCF (highest common factor) is the largest number dividing all of them; the LCM (lowest common multiple) is the smallest number they all divide into.
Both are found by prime factorisation: HCF takes the common primes to their lowest powers; LCM takes all primes to their highest powers.
The relation, for two numbers only:
HCF × LCM = the product of the two numbers
That shortcut fails for three or more, which is exactly when a question offers it.
Which one a word problem wants is the real test:
- HCF — cutting into the largest equal pieces, the largest identical groups, the biggest tile that fits exactly.
- LCM — when things happening at different intervals coincide again, the least number divisible by several, the smallest quantity that can be split evenly several ways.
Fractions and decimals
A proper fraction is less than 1, an improper fraction is 1 or more, and a mixed number pairs a whole with a proper fraction. Like fractions share a denominator; equivalent fractions have the same value in different terms.
- Comparing: bring to a common denominator, or cross-multiply. With the same numerator, the larger denominator makes the smaller fraction — the single most common misconception in primary mathematics.
- Adding and subtracting: a common denominator first. Adding numerators and denominators together is the classic error, and it is a rule imported from whole numbers.
- Multiplying: numerators together, denominators together. Note that multiplying by a proper fraction makes a number smaller — another place where whole-number intuition misleads.
- Dividing: multiply by the reciprocal.
Decimals are fractions with denominators of 10, 100, 1000. Comparing them digit by digit from the left is right; judging by the number of digits is wrong (0.5 > 0.25, though 0.25 has more digits).
Percentage is a fraction out of 100. Ratio compares two quantities of the same kind and has no units.
The order of operations
BODMAS — Brackets, Of, Division, Multiplication, Addition, Subtraction.
Division and multiplication rank equally and are worked left to right; so do addition and subtraction. That detail decides questions: 12 ÷ 3 × 2 is 8, not 2, because the division comes first from the left.
A question of the kind the paper sets
Three bells ring at intervals of 12, 15 and 18 minutes. They ring together at 9 a.m. When will they next ring together?
The bells coincide at a time that is a multiple of all three intervals, and "next" means the smallest such time — so this is LCM, not HCF.
Prime factorise: 12 = 2² × 3, 15 = 3 × 5, 18 = 2 × 3². Take every prime to its highest power: 2² × 3² × 5 = 180 minutes = 3 hours. They ring together again at 12 noon.
The distractor is 3 minutes, the HCF, offered to catch the candidate who recognised "common" and stopped reading.
Mistakes that cost marks here
- Treating 1 as a prime number.
- Confusing face value with place value.
- Saying whole numbers are closed under subtraction.
- Reaching for HCF when the question describes things coinciding, or LCM when it describes cutting into equal parts.
- Using HCF × LCM = product for three numbers. It holds for two.
- Adding fractions by adding numerators and denominators.
- Working BODMAS strictly in the order of the letters instead of left to right within each rank.
In one line
Most of the marks here turn on a handful of exact definitions — prime, factor, place value, closure — and on reading a word problem carefully enough to know whether it wants the biggest common piece or the first common moment.
For the classroom
Every rule on this page has a concrete demonstration behind it: bundles of ten for place value, paper strips for fractions, counters for factors and primes. The pedagogy paper will ask why that matters; the content paper will ask the rule. They are the same syllabus asking from two sides.
Where this comes from
Scope from the CBSE CTET Information Bulletin, Appendix I (Mathematics content: numbers, addition and subtraction, multiplication, division for Paper I; knowing our numbers, playing with numbers, whole numbers, negative numbers and integers, fractions for Paper II), and from the WBBPE TET-2023 booklet. The substance follows the NCERT mathematics books for classes III to VIII, which are the books these papers are set from. Our explanation, not a Board text.