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Number Theory

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CAT 2025 Lesson : Number Theory - Algebraic Expansion

4. Formulae and Powers to remember

4.1 Algebraic Expansions

The binomial expansion formula is provided below.

(a + b)^n =

^n C_0 a^n b^0 +

^n C_1 a^{n-1} b^1 +

^n C_2 a^{n-2} b^2 + ... +

^n C_n a^{0} b^n

Note: In the formula above,

^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

^nC_0a^nb^0

, where

r = 0

.
2) For every subsequent term,

r

increases by

1

, power of

a

reduces by

1

and power of

b

increases by

1

.
3)

\therefore

the last term is

^nC_na^0b^n

4) Note that if the power is

n

, there will be

n + 1

terms.

For instance, to derive the expansion for

(a+b)^2

, we substitute

n=2

in the formula above and get the following.

(a + b)^2 =

^2 C_0 a^2 b^0 +

^2 C_1 a^1 b^1 +

^2 C_2 a^0 b^2

⇒

(a + b)^2 = a^2 + 2ab + b^2

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 $a^3 + b^3 + c^3$ $=$	$3 abc$

The following are to be noted as well.
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.

Example 13

The remainder, when

(15^{23} + 23^{23})

is divided by

19

, is:
[CAT 2004]

(1)

4

(2)

15

(3)

0

(4)

18

Solution

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

is divisible by

(a + b)

n

is odd.

\therefore (15^{23} + 23^{23})

is divisible by

15 + 23 = 38

\therefore (15^{23} + 23^{23})

will perfectly divide

19

and leave a remainder of

0

.

Answer: (3)

0

Example 14

If R =

\dfrac{30^{65} - 29^{65}}{30^{64} + 29^{64}}

, then:
[CAT 2005]

(1)

0

< R <

0.1

(2)

0.1

< R <

0.5

(3)

0.5

< R <

1.0

(4) R >

1.0

Solution

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

30^{65} - 29^{65} = (30 - 29) (30^{64} + 30^{63} \times 29 + ... + 30 \times 29^{63} + 29^{64})

(30^{64} + 29^{64}) + (30^{63} \times 29 + ... + 30 \times 29^{63})

\therefore

R =

\dfrac{(30^{64} + 29^{64}) + (30^{63} \times 29 + ... + 30 \times 29^{63})}{30^{64} + 29^{64}}

= 1 + \dfrac{(30^{63} \times 29 + ... + 30 \times 29^{63})}{30^{64} + 29^{64}} \gt 1

Answer: (4) R >

1.0

Additionally, to improve your speed in answering questions, it is best if you memorise the following
(1) Multiplication Tables from

1

15

; and
(2) Squares and higher powers listed below provided in

4.2

and

4.3

4.2 Squares

$1^2 = 1$ $9^2 = 81$ $17^2 = 289$ $25^2 = 625$

$2^2 = 4$ $10^2 = 100$ $18^2 = 324$ $26^2 = 676$

$3^2 = 9$ $11^2 = 121$ $19^2 = 361$ $27^2 = 729$

$4^2 = 16$ $12^2 = 144$ $20^2 = 400$ $28^2 = 784$

$5^2 = 25$ $13^2 = 169$ $21^2 = 441$ $29^2 = 841$

$6^2 = 36$ $14^2 = 196$ $22^2 = 484$ $30^2 = 900$

$7^2 = 49$ $15^2 = 225$ $23^2 = 529$ $31^2 = 961$

$8^2 = 64$ $16^2 = 256$ $24^2 = 576$ $32^2 = 1024$

4.3 Higher Powers

$2^1 = 2$	$2^2 = 4$	$2^3 = 8$	$2^4 = 16$
$2^5 = 32$	$2^6 = 64$	$2^7 = 128$	$2^8 = 256$
$2^9 = 512$	$2^{10} = 1024$	$2^{11} = 2048$	$2^{12} = 4096$

$3^1 = 3$	$3^2 = 9$	$3^3 = 27$	$3^4 = 81$
$3^5 = 243$	$3^6 = 729$	$3^7 = 2187$	$3^8 = 6561$

$4^1 = 4$	$4^2 = 16$	$4^3 = 64$	$4^4 = 256$
$4^5 = 1024$	$4^6 = 4096$

$5^1 = 5$	$5^2 = 25$	$5^3 = 125$	$5^4 = 625$
$5^5 = 3125$	$5^6 = 15625$

$6^1 = 6$	$6^2 = 36$	$6^3 = 216$	$6^4 = 1296$

$7^1 = 7$	$7^2 = 49$	$7^3 = 343$	$7^4 = 2401$

$8^1 = 8$	$8^2 = 64$	$8^3 = 512$	$8^4 = 4096$

$9^1 = 9$	$9^2 = 81$	$9^3 = 729$	$9^4 = 6561$

$11^1 = 11$	$11^2 = 121$	$11^3 = 1331$	$11^4 = 14641$

$12^1 = 12$	$12^2 = 144$	$12^3 = 1728$

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$1^2 = 1$	$9^2 = 81$	$17^2 = 289$	$25^2 = 625$
$2^2 = 4$	$10^2 = 100$	$18^2 = 324$	$26^2 = 676$
$3^2 = 9$	$11^2 = 121$	$19^2 = 361$	$27^2 = 729$
$4^2 = 16$	$12^2 = 144$	$20^2 = 400$	$28^2 = 784$
$5^2 = 25$	$13^2 = 169$	$21^2 = 441$	$29^2 = 841$
$6^2 = 36$	$14^2 = 196$	$22^2 = 484$	$30^2 = 900$
$7^2 = 49$	$15^2 = 225$	$23^2 = 529$	$31^2 = 961$
$8^2 = 64$	$16^2 = 256$	$24^2 = 576$	$32^2 = 1024$