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Number Theory

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CAT 2025 Lesson : Number Theory - Concepts & Cheatsheet

Note: The video for this module contains a summary of all the concepts covered in this lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

8. Cheatsheet

1) The number system we use is the base

10

number system. It has

10

unique digits or symbols

-

0

1

2

3

4

5

6

7

8

and

9

.

2)

2

-digit number can be written as

\bm{10a + b}

, a

3

-digit number can be written as

\bm{100a + 10b + c}

, etc.

3) Sequence of natural numbers is

1

2

3

4

, ...
The sequence of whole numbers is

0

1

2

3

, ...

4) Where

n

is an integer, even numbers are of the form

2n

and odd numbers are of the form

2n + 1

.

5) Arithmetic Properties of odd and even numbers:

Addition	Multiplication
Even + Even = Even	Even $\times$ Even = Even
Even + Odd = Odd	Even $\times$ Odd = Even
Odd + Odd = Even	Odd $\times$ Odd = Odd

|x| = x

x \ge 0

\space \space \space \space \space \space \space \space\space \space = -x

x \lt 0

n! = 1 \times 2 ... \times n

Exception:

0! = 1

8) Algebraic Formulae to Memorise

Expression	Expansion
$(a+b)^{n}$	$^n C_0 a^n b^0 +$ $^n C_1 a^{n-1} b^1 + ... +$ $^nC_n a^{0} b^n$
$a^{n} - b^{n}$	$(a - b) (a^{n - 1} + a^{n - 2} b + ... + b^{n - 1})$
$(a+b)^2$	$a^2 + 2 ab + b^2$
$(a-b)^2$	$a^2 - 2 ab + b^2$
$a^2-b^2$	$(a + b) (a - b)$
$a^2+b^2$	$(a + b)^2 - 2 ab$ ; or $(a - b)^2 + 2 ab$
$(a+b)^3$	$a^3 + 3ab (a + b) + b^3$ ; or $a^3 + 3 a^2 b + 3 a b^2 + b^3$
$(a-b)^3$	$a^3 - 3ab (a - b) - b^3$ ; or $a^3 - 3 a^2 b + 3 a b^2 - b^3$
$a^3+b^3$	$(a + b) (a^2 - ab + b^2)$ ; or $(a + b)^3 - 3 ab (a + b)$
$a^3-b^3$	$(a - b) (a^2 + ab + b^2)$ ; or $(a - b)^3 + 3 ab (a - b)$
$(a + b + c)^2$	$a^2 + b^2 + c^2 + 2 ab + 2 bc + 2 ca$
$a^3 + b^3 + c^3 - 3 abc$	$(a + b + c) (a^2 + b^2 + c^2 - ab - bc - ca)$
If $a+b+c=0$ , then $a^3 + b^3 + c^3$ $=$	$3 abc$

Note that
1)

(a^{n} + b^{n})

is divisible by

(a + b)

n

is odd.
2)

(a^{n} - b^{n})

is divisible by

(a + b)

n

is even.

9) Properties of Prime and Composite numbers

(a) Prime Numbers are natural numbers that have exactly

2

factors:

1

and the number itself.
(b) Composite numbers are natural numbers that have more than

2

factors.
(c)

1

is the only natural number that is neither prime nor composite.
(d)

2

is the smallest prime and the only even prime.
(e) All prime numbers except

2

and

5

end with the digits

1

3

7

9

.
(f) There are

15

prime numbers less than

50

and

25

prime numbers less than

100

.

10) A number is prime if none of the prime numbers less than its square root divides it.

11) Two numbers are coprime if they do not have any common factors.

12) If

x

and

y

are integers with

y \gt x

, the number of integers between

y

and

x

Condition	Number of Integers
where $y$ and $x$ are both to be included	$y - x + 1$
where one of $x$ and $y$ is to be included while the other is not	$y - x$
where $y$ and $x$ are both to be excluded	$y - x - 1$

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