Number types, place value & counting — notes, question types & practice · Number System

01

Overview

Every number in the exam belongs to one or more families. This lesson covers the families, prime and co-prime numbers, place value, counting multiples in a range, sums of standard series, page numbering, and reversed two-digit numbers. These are quick one-mark questions, so speed matters most.

02

The number families

Family	Members	Watch point
Natural	$1, 2, 3, \ldots$	$0$ is not natural
Whole	$0, 1, 2, \ldots$	smallest whole number is $0$
Integers	$\ldots, -2, -1, 0, 1, 2, \ldots$	$0$ is neither positive nor negative
Rational	$\dfrac{p}{q}$ with $q \ne 0$	terminating and repeating decimals qualify
Irrational	never ends, never repeats	$\sqrt{2}$ and $\pi$ ; sums can still be rational
Real	rational and irrational together	every point on the number line

Watch: $0$ is whole but not natural, and $1$ is neither prime nor composite. Both lines appear as free marks.

03

Prime, composite and co-prime

A prime has exactly two factors: $1$ and itself. A composite number has more than two factors. $2$ is the only even prime, and there are $25$ primes below $100$ . Co-prime numbers share only the factor $1$ : $8$ and $15$ are co-prime though neither is prime. Consecutive integers are always co-prime.

To test whether a number is prime, divide it by every prime up to its square root. For $221$ the square root is below $15$ , and $221 = 13 \times 17$ , so it is composite.

04

Place value and face value

The face value of a digit is the digit itself. The place value is the digit times the value of its position. In $5,63,847$ the digit $6$ sits in the ten-thousands place, so its place value is $6 \times 10000 = 60000$ while its face value is $6$ .

Rule: Place value $-$ face value $=$ digit $\times$ (position value $- 1$ ). For $8$ in $9,28,475$ : $8 \times 9999 = 79992$ .

05

Counting multiples in a range

Multiples of $k$ up to $n$ number exactly the whole-number part of $n \div k$ . For a range, count up to the end and subtract the count before the start.

Rule: Multiples of $k$ from $a$ to $b$ = $\left\lfloor \dfrac{b}{k} \right\rfloor - \left\lfloor \dfrac{a-1}{k} \right\rfloor$ . Multiples of $7$ from $100$ to $300$ : $42 - 14 = 28$ .

Read the joining word with care. "Divisible by $4$ and $6$ " means divisible by $\mathrm{LCM}(4, 6) = 12$ , not by $24$ . "Divisible by $4$ or $6$ " adds the two counts and subtracts the multiples of $12$ . "Neither" takes the total minus the "or" count.

06

Sums of common series

Series	Sum
$1 + 2 + \cdots + n$	$\dfrac{n(n+1)}{2}$
$1^2 + 2^2 + \cdots + n^2$	$\dfrac{n(n+1)(2n+1)}{6}$
$1^3 + 2^3 + \cdots + n^3$	$\left[\dfrac{n(n+1)}{2}\right]^2$
first $n$ odd numbers	$n^2$
first $n$ even numbers	$n(n+1)$

For a sum that starts late, find the full sum and subtract the missing head. For $11^2$ to $20^2$ : the sum up to $20$ is $2870$ , the sum up to $10$ is $385$ , and the answer is $2485$ .

Tip: Any evenly spaced list sums to: number of terms $\times$ (first $+$ last) $\div 2$ . Evens from $12$ to $48$ : $19$ terms, sum $19 \times 30 = 570$ .

07

Digits used in page numbering

Pages $1$ to $9$ use $9$ digits. Pages $10$ to $99$ use $90 \times 2 = 180$ digits. Each later page uses as many digits as it has. For $350$ pages: $9 + 180 + 251 \times 3 = 942$ digits. The reverse question subtracts $189$ , divides by $3$ , and adds $99$ to the quotient.

08

Two-digit numbers and their reverse

Write the number as $10a + b$ ; its reverse is $10b + a$ . Their difference is $9(a - b)$ and their sum is $11(a + b)$ . A number that exceeds its reverse by $27$ has digits differing by $27 \div 9 = 3$ . For example $74$ and $47$ differ by $27$ .

09

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Counting numbers divisible by k in a range

How to spot it:

A range is given and the question asks how many numbers in it are divisible by one or two given numbers. Words like and, or, neither or but not decide the method.

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

Method

For one divisor $k$ : count multiples up to the end, then subtract the multiples before the start.
For 'by $x$ and $y$ ': count with $k = \mathrm{LCM}(x, y)$ instead of $x \times y$ .
For 'by $x$ or $y$ ': add both counts, then subtract the count of the LCM once.
For 'neither': subtract the 'or' count from the size of the range.

Why it works:

Every k-th number is a multiple of k, so floor division counts them without any listing.

Try this

How many numbers from 1 to 300 are divisible by 4 or 6?

Show solution

By 4: $300 \div 4$ gives 75. By 6: $300 \div 6$ gives 50.
Both: multiples of $\mathrm{LCM}(4, 6) = 12$ , and $300 \div 12 = 25$ .
$75 + 50 - 25 = 100$ .

Answer

100

Type 2very common3 practice Q

Sum of a series or of multiples in a range

How to spot it:

The question asks for a sum: all two-digit multiples of a number, squares from one point to another, or the first n odd numbers.

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

Method

Count the terms: for multiples of $d$ from $a$ to $l$ , the count is $\dfrac{l - a}{d} + 1$ .
Add the first and last terms and halve the total to get the average.
Multiply: sum $=$ count $\times$ average.
For squares or cubes, take the full sum formula and subtract the head you do not need.

Why it works:

Pairing first with last, second with second-last, and so on gives equal totals, so the sum is the count times the average.

Try this

Find the sum of all two-digit multiples of 7.

Show solution

First two-digit multiple: $14$ . Last: $98$ .
Count: $\dfrac{98 - 14}{7} + 1 = 13$ terms.
Sum $= \dfrac{13 \times (14 + 98)}{2} = 13 \times 56 = 728$ .

Answer

728

Type 3common2 practice Q

Place value and face value

How to spot it:

A number is written with Indian commas and the question asks for the place value, the face value, or a combination of both for one digit.

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

Method

Mark positions from the right: ones, tens, hundreds, thousands, ten-thousands, lakhs.
Place value $=$ digit $\times$ that position's value. Face value $=$ the digit itself.
Do the asked operation: sum, difference or product.

Why it works:

Our system is positional, so the same digit is worth ten times more in each step to the left.

Try this

Find the difference between the place value and the face value of 5 in 8,45,327.

Show solution

Digit $5$ is in the thousands place: place value $= 5 \times 1000 = 5000$ .
Face value $= 5$ .
Difference $= 5000 - 5 = 4995$ .

Answer

4995

Type 4common3 practice Q

Classifying numbers: prime, rational, co-prime

How to spot it:

Options list numbers and the question asks which one is prime, rational, irrational, or co-prime with another.

Method

Test primes by dividing by every prime up to the square root of the number.
Simplify before classifying: a product of surds can collapse to a rational number.
Co-prime just needs HCF 1; neither number has to be prime.

Why it works:

A number is irrational only if its simplified value still has a decimal that never ends and never repeats.

Try this

Is 407 a prime number?

Show solution

$\sqrt{407}$ is just above $20$ , so test primes up to $20$ .
$407 = 11 \times 37$ .
Two factors besides 1, so it is composite.

Answer

No, 407 = 11 x 37

Type 5common3 practice Q

Digits used in page numbering

How to spot it:

The question asks how many digits a book of N pages needs, or gives the digit count and asks for the pages.

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

Method

Pages 1 to 9 use 9 digits at one digit each.
Pages 10 to 99 use $90 \times 2 = 180$ digits.
Pages beyond 99 use one digit per digit of the page number; multiply and add.
For the reverse question, strip 9 and 180 first, then divide what remains.

Why it works:

Each block of page numbers has a fixed digit length, so the count splits into three clean parts.

Try this

How many digits are used to number the pages of a 200-page book?

Show solution

Single-digit pages: $9$ digits.
Two-digit pages: $90 \times 2 = 180$ digits.
Pages 100 to 200: $101 \times 3 = 303$ digits.
Total: $9 + 180 + 303 = 492$ .

Answer

492

Type 6very common3 practice Q

Two-digit numbers and digit reversal

How to spot it:

The digits of a two-digit number are reversed and the number changes by a known amount, or consecutive numbers are summed.

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

Method

Write the number as $10a + b$ with $a$ the tens digit and $b$ the units digit.
A reversal difference divided by 9 gives the digit difference $a - b$ .
A reversal sum divided by 11 gives the digit sum $a + b$ .
For consecutive numbers, divide the sum by the count to get the middle number.

Why it works:

The tens digit is worth ten times the units digit, so a swap moves 9 per unit of digit difference.

Try this

A two-digit number exceeds the number formed by reversing its digits by 27. What is the difference between its digits?

Show solution

Difference $= 9(a - b) = 27$ .
$a - b = 27 \div 9 = 3$ .
Example: $74 - 47 = 27$ .

Answer

3

10

Formula sheet

Multiples of k up to n

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

Multiples of k from a to b

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

Inclusion-exclusion

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

the overlap holds multiples of the LCM

Sum of first n natural numbers

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

Sum of squares

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

Sum of cubes

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

First n odd or even numbers

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

Any arithmetic series

S = \dfrac{n}{2}(a + l)

terms times average of first and last

Digit reversal

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

11

Shortcuts that save time

⚡ Floor-division counting

Count multiples of k up to the end of the range, then subtract the multiples before the start. No listing needed.

Example

How many numbers from 100 to 500 are divisible by 17?

Show solution

Up to $500$ : $\left\lfloor \dfrac{500}{17} \right\rfloor = 29$ .
Below $100$ : $\left\lfloor \dfrac{99}{17} \right\rfloor = 5$ .
Answer: $29 - 5 = 24$ .

Answer

24

⚡ Place value in one product

Place value minus face value equals digit times (position value minus 1). The 9999 pattern does it in one line.

Example

Find the difference between the place value and the face value of 8 in 9,28,475.

Show solution

Digit $8$ sits in the ten-thousands place: place value $= 8 \times 10000 = 80000$ .
Difference $= 8 \times (10000 - 1) = 8 \times 9999 = 79992$ .

Answer

79992

⚡ Middle term times count

For any evenly spaced list, the sum equals the number of terms times the middle term. The middle term is the average of first and last.

Example

Find the sum of all even numbers from 12 to 48.

Show solution

Count: $\dfrac{48 - 12}{2} + 1 = 19$ terms.
Average $= \dfrac{12 + 48}{2} = 30$ .
Sum $= 19 \times 30 = 570$ .

Answer

570

12

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Counting the end points when the question says between.

Between 1 and 100 means 2 to 99. From 1 to 100 includes both ends.

Mistake 02

Treating 1 as a prime number.

1 is neither prime nor composite, and 2 is the only even prime.

Mistake 03

In 'divisible by 3 or 5' questions, adding both counts and stopping.

Add both counts, then subtract the multiples of 15 once.

Mistake 04

Using the sum formula for a series that starts away from 1.

Find the full sum up to the end, then subtract the sum before the start.

Mistake 05

Reading 'divisible by 4 and 6' as divisible by 24.

It means divisible by LCM(4, 6) = 12. Join 'and' divisors through their LCM.

13

Quick revision

Read this the night before the exam.

Multiples of $k$ from $a$ to $b$ : $\left\lfloor \dfrac{b}{k} \right\rfloor - \left\lfloor \dfrac{a-1}{k} \right\rfloor$ .
'And' means the LCM; 'or' means add counts and subtract the LCM count.
Sum formulas: $\dfrac{n(n+1)}{2}$ , $\dfrac{n(n+1)(2n+1)}{6}$ , $\left[\dfrac{n(n+1)}{2}\right]^2$ , $n^2$ for odd numbers.
Place value $-$ face value $=$ digit $\times$ (position value $- 1$ ).
Reversal: difference $= 9(a-b)$ , sum $= 11(a+b)$ .
$0$ is whole but not natural; $1$ is neither prime nor composite.

14

Practice: 19 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

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