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Number Theory

Number Theory

MODULES

Basics of Numbers
Types of Numbers
Fractions
Arithmetic Operations
Other Numerical Operations
Algebraic Expansion
Prime Numbers
Counting Integers
Past Questions

CONCEPTS & CHEATSHEET

Concept Revision Video

SPEED CONCEPTS

Number Theory 1
-/10
Number Theory 2
-/10
Algebraic Expansion 1
-/10

PRACTICE

Number Theory : Level 1
Number Theory : Level 2
Number Theory : Level 3
ALL MODULES

CAT 2025 Lesson : Number Theory - Concepts & Cheatsheet

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Note: The video for this module contains a summary of all the concepts covered in the Number Theory 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 101010 number system. It has 101010 unique digits or symbols −-− 000, 111, 222, 333, 444, 555, 666, 777, 888 and 999.

2)
222-digit number can be written as 10a+b\bm{10a + b}10a+b, a 333-digit number can be written as 100a+10b+c\bm{100a + 10b + c}100a+10b+c, etc.

3) Sequence of natural numbers is
111, 222, 333, 444, ...
The sequence of whole numbers is
000, 111, 222, 333, ...

4) Where
nnn is an integer, even numbers are of the form 2n2n2n and odd numbers are of the form 2n+12n + 12n+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

6)
∣x∣=x|x| = x∣x∣=x if x≥0x \ge 0x≥0
          =−x\space \space \space \space \space \space \space \space\space \space = -x          =−x if x<0x \lt 0x<0

7)
n!=1×2...×nn! = 1 \times 2 ... \times nn!=1×2...×n

Exception:
0!=10! = 10!=1

8) Algebraic Formulae to Memorise

Expression Expansion
(a+b)n(a+b)^{n}(a+b)n nC0anb0+^n C_0 a^n b^0 +nC0​anb0+ nC1an−1b1+...+^n C_1 a^{n-1} b^1 + ... +nC1​an−1b1+...+ nCna0bn^nC_n a^{0} b^nnCn​a0bn
an−bna^{n} - b^{n}an−bn (a−b)(an−1+an−2b+...+bn−1)(a - b) (a^{n - 1} + a^{n - 2} b + ... + b^{n - 1})(a−b)(an−1+an−2b+...+bn−1)
(a+b)2(a+b)^2(a+b)2 a2+2ab+b2a^2 + 2 ab + b^2a2+2ab+b2
(a−b)2(a-b)^2(a−b)2 a2−2ab+b2a^2 - 2 ab + b^2a2−2ab+b2
a2−b2a^2-b^2a2−b2 (a+b)(a−b)(a + b) (a - b)(a+b)(a−b)
a2+b2a^2+b^2a2+b2 (a+b)2−2ab(a + b)^2 - 2 ab(a+b)2−2ab; or
(a−b)2+2ab(a - b)^2 + 2 ab(a−b)2+2ab
(a+b)3(a+b)^3(a+b)3 a3+3ab(a+b)+b3a^3 + 3ab (a + b) + b^3a3+3ab(a+b)+b3; or
a3+3a2b+3ab2+b3a^3 + 3 a^2 b + 3 a b^2 + b^3a3+3a2b+3ab2+b3
(a−b)3(a-b)^3(a−b)3 a3−3ab(a−b)−b3a^3 - 3ab (a - b) - b^3a3−3ab(a−b)−b3; or
a3−3a2b+3ab2−b3a^3 - 3 a^2 b + 3 a b^2 - b^3a3−3a2b+3ab2−b3
a3+b3a^3+b^3a3+b3 (a+b)(a2−ab+b2)(a + b) (a^2 - ab + b^2) (a+b)(a2−ab+b2); or
(a+b)3−3ab(a+b)(a + b)^3 - 3 ab (a + b)(a+b)3−3ab(a+b)
a3−b3a^3-b^3a3−b3 (a−b)(a2+ab+b2)(a - b) (a^2 + ab + b^2) (a−b)(a2+ab+b2); or
(a−b)3+3ab(a−b)(a - b)^3 + 3 ab (a - b)(a−b)3+3ab(a−b)
(a+b+c)2(a + b + c)^2(a+b+c)2 a2+b2+c2+2ab+2bc+2caa^2 + b^2 + c^2 + 2 ab + 2 bc + 2 caa2+b2+c2+2ab+2bc+2ca
a3+b3+c3−3abca^3 + b^3 + c^3 - 3 abca3+b3+c3−3abc (a+b+c)(a2+b2+c2−ab−bc−ca)(a + b + c) (a^2 + b^2 + c^2 - ab - bc - ca)(a+b+c)(a2+b2+c2−ab−bc−ca)
If a+b+c=0a+b+c=0a+b+c=0, then
a3+b3+c3a^3 + b^3 + c^3a3+b3+c3 ===
3abc3 abc3abc

Note that
1)
(an+bn)(a^{n} + b^{n})(an+bn) is divisible by (a+b)(a + b)(a+b) if nnn is odd.
2)
(an−bn)(a^{n} - b^{n})(an−bn) is divisible by (a+b)(a + b)(a+b) if nnn is even.

9) Properties of Prime and Composite numbers

(a) Prime Numbers are natural numbers that have exactly
222 factors: 111 and the number itself.
(b) Composite numbers are natural numbers that have more than
222 factors.
(c)
111 is the only natural number that is neither prime nor composite.
(d)
222 is the smallest prime and the only even prime.
(e) All prime numbers except
222 and 555 end with the digits 111, 333, 777 or 999.
(f) There are
151515 prime numbers less than 505050 and 252525 prime numbers less than 100100100.

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
xxx and yyy are integers with y>xy \gt xy>x, the number of integers between yyy and xxx

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

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