calendarBack
Quant

/

Numbers

/

Number Theory
ALL MODULES

CAT 2025 Lesson : Number Theory - Algebraic Expansion

bookmarked

4. Formulae and Powers to remember

4.1 Algebraic Expansions

The binomial expansion formula is provided below.

(a+b)n=(a + b)^n = nC0anb0+^n C_0 a^n b^0 + nC1an−1b1+^n C_1 a^{n-1} b^1 + nC2an−2b2+...+^n C_2 a^{n-2} b^2 + ... + nCna0bn^n C_n a^{0} b^n

Note: In the formula above, nCr=n!r! (n−r)!^nC_{r} = \dfrac{n!}{r! \space (n - r)!}. This is detailed in Permutations and Combinations lesson.

It is easy to remember this if you observe the following.

1) The first term is nC0anb0^nC_0a^nb^0, where r=0r = 0.
2) For every subsequent term, rr increases by 11, power of aa reduces by 11 and power of bb increases by 11.
3) ∴\therefore the last term is nCna0bn^nC_na^0b^n
4) Note that if the power is nn, there will be n+1n + 1 terms.

For instance, to derive the expansion for (a+b)2(a+b)^2, we substitute n=2n=2 in the formula above and get the following.

(a+b)2=(a + b)^2 = 2C0a2b0+^2 C_0 a^2 b^0 + 2C1a1b1+^2 C_1 a^1 b^1 + 2C2a0b2^2 C_2 a^0 b^2

⇒ (a+b)2=a2+2ab+b2 (a + b)^2 = a^2 + 2ab + b^2

Algebraic Formulae to Memorise

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


The following are to be noted as well.
1) (an+bn)(a^{n} + b^{n}) is divisible by (a+b)(a + b) if nn is odd.
2) (an−bn)(a^{n} - b^{n}) is divisible by (a+b)(a + b) if nn is even.

Example 13

The remainder, when (1523+2323)(15^{23} + 23^{23}) is divided by 1919, is:
[CAT 2004]

(1) 44            (2) 1515            (3) 00            (4) 1818

Solution

(an+bn)(a^{n} + b^{n}) is divisible by (a+b)(a + b) if nn is odd.

∴(1523+2323)\therefore (15^{23} + 23^{23}) is divisible by 15+23=3815 + 23 = 38

∴(1523+2323)\therefore (15^{23} + 23^{23}) will perfectly divide 1919 and leave a remainder of 00.

Answer: (3) 00


Example 14

If R = 3065−29653064+2964\dfrac{30^{65} - 29^{65}}{30^{64} + 29^{64}}, then:
[CAT 2005]

(1) 00 < R < 0.10.1            (2) 0.10.1 < R < 0.50.5            (3) 0.50.5 < R < 1.01.0            (4) R > 1.01.0

Solution

an−bn=(a−b)(an−1+an−2b+...+bn−1)a^{n} - b^{n} = (a - b) (a^{n - 1} + a^{n - 2} b + ... + b^{n - 1})

3065−2965=(30−29)(3064+3063×29+...+30×2963+2964)30^{65} - 29^{65} = (30 - 29) (30^{64} + 30^{63} \times 29 + ... + 30 \times 29^{63} + 29^{64})

= (3064+2964)+(3063×29+...+30×2963)(30^{64} + 29^{64}) + (30^{63} \times 29 + ... + 30 \times 29^{63})

∴\therefore R = (3064+2964)+(3063×29+...+30×2963)3064+2964\dfrac{(30^{64} + 29^{64}) + (30^{63} \times 29 + ... + 30 \times 29^{63})}{30^{64} + 29^{64}}

=1+(3063×29+...+30×2963)3064+2964>1= 1 + \dfrac{(30^{63} \times 29 + ... + 30 \times 29^{63})}{30^{64} + 29^{64}} \gt 1

Answer: (4) R > 1.01.0

Additionally, to improve your speed in answering questions, it is best if you memorise the following
(1) Multiplication Tables from 11 to 1515; and
(2) Squares and higher powers listed below provided in 4.24.2 and 4.34.3.

4.2 Squares

12=11^2 = 1 92=819^2 = 81 172=28917^2 = 289 252=62525^2 = 625
22=42^2 = 4 102=10010^2 = 100 182=32418^2 = 324 262=67626^2 = 676
32=93^2 = 9 112=12111^2 = 121 192=36119^2 = 361 272=72927^2 = 729
42=164^2 = 16 122=14412^2 = 144 202=40020^2 = 400 282=78428^2 = 784
52=255^2 = 25 132=16913^2 = 169 212=44121^2 = 441 292=84129^2 = 841
62=366^2 = 36 142=19614^2 = 196 222=48422^2 = 484 302=90030^2 = 900
72=497^2 = 49 152=22515^2 = 225 232=52923^2 = 529 312=96131^2 = 961
82=648^2 = 64 162=25616^2 = 256 242=57624^2 = 576 322=102432^2 = 1024

4.3 Higher Powers

21=22^1 = 2 22=42^2 = 4 23=82^3 = 8 24=162^4 = 16
25=322^5 = 32 26=642^6 = 64 27=1282^7 = 128 28=2562^8 = 256
29=5122^9 = 512 210=10242^{10} = 1024 211=20482^{11} = 2048 212=40962^{12} = 4096
31=33^1 = 3 32=93^2 = 9 33=273^3 = 27 34=813^4 = 81
35=2433^5 = 243 36=7293^6 = 729 37=21873^7 = 2187 38=65613^8 = 6561
41=44^1 = 4 42=164^2 = 16 43=644^3 = 64 44=2564^4 = 256
45=10244^5 = 1024 46=40964^6 = 4096
51=55^1 = 5 52=255^2 = 25 53=1255^3 = 125 54=6255^4 = 625
55=31255^5 = 3125 56=156255^6 = 15625
61=66^1 = 6 62=366^2 = 36 63=2166^3 = 216 64=12966^4 = 1296
71=77^1 = 7 72=497^2 = 49 73=3437^3 = 343 74=24017^4 = 2401
81=88^1 = 8 82=648^2 = 64 83=5128^3 = 512 84=40968^4 = 4096
91=99^1 = 9 92=819^2 = 81 93=7299^3 = 729 94=65619^4 = 6561
111=1111^1 = 11 112=12111^2 = 121 113=133111^3 = 1331 114=1464111^4 = 14641
121=1212^1 = 12 122=14412^2 = 144 123=172812^3 = 1728

Want to read the full content

Unlock this content & enjoy all the features of the platform

Subscribe Now arrow-right
videovideo-lock