Number System

high importance~2 Q in Tier 133 formulas⚡ 18 shortcuts6 subtopics

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Number types, place value & counting

Number families you must know cold

Type	Meaning	Watch-out
Natural	1, 2, 3, …	0 is not natural
Whole	0, 1, 2, …	smallest whole number is 0
Integers	…, −2, −1, 0, 1, 2, …	0 is neither positive nor negative
Rational	$\frac{p}{q}$ , $q \neq 0$	every terminating/recurring decimal is rational
Irrational	non-terminating, non-recurring	$\sqrt{2}$ , $\pi$ ; sum of two irrationals can be rational
Prime	exactly two factors	1 is neither prime nor composite; 2 is the only even prime; 25 primes up to 100
Co-prime	HCF = 1	8 and 15 are co-prime though neither is prime

Place value vs face value: in 3,47,218 the face value of 7 is 7, the place value is 7000.

Counting multiples in a range and sums of series (natural numbers, squares, cubes, odd numbers, AP) appear as 1-mark sitters in Tier 1. Always read between (exclusive) vs from … to (inclusive).

📖 Notes 🧩 6 question types ⚡ Formulas 🎯 19 practice Q

Detailed notes

What is a number system?

A number system is simply the family tree of numbers. Every number you meet in an exam belongs to one or more families. Knowing the families helps you answer "which of these is rational / prime / whole?" questions in seconds.

The number families

Natural numbers: counting numbers 1, 2, 3, … (0 is not natural).
Whole numbers: 0, 1, 2, 3, … (natural numbers plus 0).
Integers: …, −3, −2, −1, 0, 1, 2, 3, … 0 is neither positive nor negative.
Rational numbers: can be written as $\frac{p}{q}$ with integers $p, q$ and $q \neq 0$ . Every terminating decimal (0.25) and every recurring decimal ( $0.\overline{3}$ ) is rational.
Irrational numbers: decimal never ends and never repeats, e.g. $\sqrt{2}$ , $\sqrt{3}$ , $\pi$ . Careful: $\sqrt{8} \times \sqrt{2} = 4$ is rational, and $(2+\sqrt{3}) + (2-\sqrt{3}) = 4$ is rational.
Real numbers: rational + irrational together.

Prime, composite and co-prime

A prime has exactly two factors: 1 and itself (2, 3, 5, 7, 11, …). 2 is the only even prime.
A composite number has more than two factors (4, 6, 9, …).
1 is neither prime nor composite.
There are 25 primes up to 100 (15 up to 50, 10 from 51 to 100).
Co-prime numbers have HCF 1, e.g. 8 and 15. They need not be prime themselves.
To test if a number $N$ is prime, divide it by primes up to $\sqrt{N}$ . Example: 221 → $\sqrt{221} \approx 14.9$ ; 221 = 13 × 17, so not prime.

Place value and face value

In 5,63,847 the face value of 6 is just 6. Its place value is 6 × 10,000 = 60,000 because it sits in the ten-thousands place. Place value − face value = digit × (position value − 1).

Counting multiples in a range

Multiples of $k$ from 1 to $n$ = $\lfloor n/k \rfloor$ (whole-number part of the division). Multiples of $k$ from $a$ to $b$ = $\lfloor b/k \rfloor - \lfloor (a-1)/k \rfloor$ . Example: multiples of 7 from 100 to 300 = 42 − 14 = 28.

"Divisible by 4 and 6" means divisible by LCM(4, 6) = 12 (not 24).
"Divisible by 4 or 6" uses inclusion–exclusion: $n(A) + n(B) - n(A \cap B)$ .
"Divisible by neither" = total − (divisible by 4 or 6).
Read the words: "between 1 and 100" usually excludes the ends; "from 1 to 100" includes them.

Sums you must know

Series	Sum
1 + 2 + … + n	$\frac{n(n+1)}{2}$
$1^2 + 2^2 + … + n^2$	$\frac{n(n+1)(2n+1)}{6}$
$1^3 + 2^3 + … + n^3$	$\left[\frac{n(n+1)}{2}\right]^2$
first n odd numbers	$n^2$
first n even numbers	$n(n+1)$
any AP	number of terms × (first + last) ÷ 2

For a sum that starts in the middle (like $11^2$ to $20^2$ ), find the sum up to 20 and subtract the sum up to 10.

Digits and pages

To number pages 1 to 350: 1-digit pages 1–9 use 9 digits, 2-digit pages 10–99 use 90 × 2 = 180 digits, and pages 100–350 use 251 × 3 = 753 digits. Total 942. Going backwards (digits given, find pages) just reverses these steps.

Two-digit numbers and reversal

A two-digit number with tens digit $a$ and units digit $b$ is $10a + b$ . Its reverse is $10b + a$ .

Difference = $9(a - b)$ . Sum = $11(a + b)$ .
So "number − reverse = 36" tells you the digits differ by 4 straight away.

Quick revision

0 is whole but not natural; 1 is neither prime nor composite; 2 is the only even prime.
25 primes below 100.
Count multiples with floor division; "and" means LCM, "or" means inclusion–exclusion.
Sum formulas: $\frac{n(n+1)}{2}$ , $\frac{n(n+1)(2n+1)}{6}$ , $\left[\frac{n(n+1)}{2}\right]^2$ , $n^2$ .
Reversal: difference = 9 × (digit difference), sum = 11 × (digit sum).

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Counting numbers divisible by k in a range (and / or / neither)very common3 practice Q

How to spot it:

The question asks 'how many numbers between/from A to B are divisible by …'. Words like 'and', 'or', 'neither', 'but not' decide the method.

\left\lfloor \frac{b}{k} \right\rfloor - \left\lfloor \frac{a-1}{k} \right\rfloor

For one divisor $k$ : count up to $b$ , subtract the count up to $a-1$ .
"Divisible by $x$ and $y$ " → use $k = \text{LCM}(x, y)$ .
"By $x$ or $y$ " → $n(x) + n(y) - n(\text{LCM})$ .
"Neither" → total − (or-count). "By $x$ but not $y$ " → $n(x) - n(\text{LCM})$ .

Why it works: every $k$ -th number is a multiple of $k$ , so floor division counts them without listing.

Example: How many numbers from 1 to 200 are divisible by 6 or 8?

By 6: $\lfloor 200/6 \rfloor = 33$ . By 8: $\lfloor 200/8 \rfloor = 25$ . By LCM 24: $\lfloor 200/24 \rfloor = 8$ . Answer $= 33 + 25 - 8 = 50$ .

Type 2: Sum of series: natural numbers, squares, cubes, odd numbers, APvery common3 practice Q

How to spot it:

A long sum like $1^2 + 2^2 + \dots$ , 'sum of all two-digit multiples of 7', or 'sum of first n odd numbers'.

\frac{n(n+1)}{2},\ \frac{n(n+1)(2n+1)}{6},\ \left[\frac{n(n+1)}{2}\right]^2,\ S_{AP} = \frac{n}{2}(a+l)

Identify the series type.
If it does not start at 1, take (sum up to the last term) − (sum up to the term just before the first).
For multiples/AP: number of terms $n = \frac{l-a}{d} + 1$ , then sum $= \frac{n}{2}(a + l)$ .

Why it works: an AP pairs up first-and-last, second-and-second-last … each pair has the same total.

Example: Find $1^3 + 2^3 + \dots + 10^3$ .

$\left[\frac{10 \times 11}{2}\right]^2 = 55^2 = 3025$ .

Type 3: Place value and face valuecommon2 practice Q

How to spot it:

A number is given in Indian commas and the question asks the place value, face value, their sum/difference/product for one or two digits.

\text{Place value} = \text{digit} \times 10^{\text{position}}

Mark positions from the right: ones, tens, hundreds, thousands, ten-thousands, lakhs, ten-lakhs, crores.
Place value = digit × that position's value. Face value = the digit itself.
Do the asked operation.

Why: our system is positional — the same digit is worth more as it moves left.

Example: Find the difference between the place value and the face value of 4 in 7,64,219.

4 is in the thousands place → place value 4000. Face value 4. Difference $= 4000 - 4 = 3996$ .

Type 4: Classifying numbers: prime, rational/irrational, co-primecommon3 practice Q

How to spot it:

Options list numbers and the question asks which is prime / rational / irrational, or asks how many primes lie in a range.

For rational vs irrational, simplify first ( $\sqrt{8}\times\sqrt{2} = 4$ is rational).
For primes, test divisibility by primes up to $\sqrt{N}$ .
Remember: 1 is not prime; 2 is the only even prime; 25 primes up to 100.

Why: a number is irrational only if its simplified value still has a non-ending, non-repeating decimal.

Example: Is 391 a prime number?

$\sqrt{391} \approx 19.8$ , so test primes up to 19. $391 = 17 \times 23$ . So 391 is not prime.

Type 5: Digits used in page numbering / counting a digitcommon3 practice Q

How to spot it:

'How many digits are needed to number a book of N pages?' or the reverse, or 'how many times does digit 5 appear from 1 to 100?'

\text{Digits} = 9 + 180 + 3(N - 99) \quad (100 \le N \le 999)

Pages 1–9 use 9 digits, pages 10–99 use 180 digits, each 3-digit page uses 3 digits.
Reverse problem: subtract 189, divide the rest by 3, add 99.
For counting a digit $d$ (not 0) from 1 to 99: it appears 10 times in the units place and 10 times in the tens place = 20.

Why: each group of numbers has a fixed number of digits.

Example: How many digits are used to number the pages of a 125-page book?

$9 + 180 + 3 \times (125 - 99) = 9 + 180 + 78 = 267$ .

Type 6: Two-digit numbers, digit reversal and consecutive numbersvery common3 practice Q

How to spot it:

'If the digits are reversed, the number increases by 27', 'sum of three consecutive even numbers is …'.

(10a+b) - (10b+a) = 9(a-b),\quad (10a+b) + (10b+a) = 11(a+b)

Write the number as $10a + b$ .
Reversal difference gives $a - b$ (divide by 9); reversal sum gives $a + b$ (divide by 11).
Consecutive numbers: take the middle one as $x$ (for evens/odds the neighbours are $x \pm 2$ ). Sum ÷ count = middle number.
Or simply test the options.

Why: the tens digit is worth 10 times the units digit, so swapping them changes the value by 9 per unit of difference.

Example: The sum of five consecutive odd numbers is 135. Find the largest.

Middle number $= 135 \div 5 = 27$ . The numbers are 23, 25, 27, 29, 31. Largest $= 31$ .

Formulas

Multiples of k from 1 to n

\left\lfloor \frac{n}{k} \right\rfloor

Multiples of k from a to b

\left\lfloor \frac{b}{k} \right\rfloor - \left\lfloor \frac{a-1}{k} \right\rfloor

Inclusion–exclusion

n(A \cup B) = n(A) + n(B) - n(A \cap B)

A ∩ B = multiples of LCM

Sum of first n natural numbers

\frac{n(n+1)}{2}

Sum of squares

1^2+2^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}

Sum of cubes

1^3+2^3+\cdots+n^3 = \left[\frac{n(n+1)}{2}\right]^2

First n odd / even numbers

1+3+\cdots+(2n-1) = n^2,\quad 2+4+\cdots+2n = n(n+1)

AP: terms and sum

n = \frac{l-a}{d}+1,\quad S = \frac{n}{2}(a+l)

Shortcut tricks

⚡ Floor-division counting

Count multiples of $k$ up to $b$ , subtract the multiples up to $a-1$ . No listing needed.

Example: How many numbers from 100 to 400 are divisible by 13?

$\left\lfloor \frac{400}{13} \right\rfloor - \left\lfloor \frac{99}{13} \right\rfloor = 30 - 7 = 23$ .

⚡ Place value − face value

Place value = digit × its position value. Difference = digit × (position value − 1).

Example: Find the difference between the place value and face value of 7 in 3,47,218.

Place value $= 7000$ , face value $= 7$ . Difference $= 7 \times 999 = 6993$ .

⚡ AP sum = terms × middle term

For any AP, sum = number of terms × average of first and last term. Saves writing the formula.

Example: Find the sum of all odd numbers from 21 to 79.

Terms $= \frac{79-21}{2}+1 = 30$ , average $= \frac{21+79}{2} = 50$ . Sum $= 30 \times 50 = 1500$ .

Where students lose marks

Counting the end points when the question says between (exclusive).
Treating 1 as a prime number, or forgetting that 2 is prime.
In 'divisible by 3 or 5' questions, adding both counts without subtracting multiples of 15.
Using $\frac{n(n+1)}{2}$ for a series that does not start at 1.

Practice sets — 19 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 10 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.