Class 12 Mathematics solutions, Study notes of Mathematics

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Exose-

sa Euulid'a diwiaion algrilhum do Jund Jha

HCF

135 and 225

nlution a>b^ a^225 Ad^ b=

a =btM

3 12p

225 135 XI +

135 90 X^ 1tH

901355

90 M 2 +O

2

Hau H=0 HCF is 5 45)

i 136 and 38 220

daluion ab^ a^38220 and^ b 136

a

=

bq t

38220 134]x^

195 +0 (^196) J#h

6 He = 0

HCF (^) o 196 and (^38220 )

oc9 80

l 86t and 255

8oluton- ayb a = 86 and b= 255

a b t

25s) 8

86 255 X3 to

102. 255

20 4

255= 102 X2 t5||

2ST

5i)

102 sx 2+

Hete F

.', HCF

86 and^255 S

. dn^ army (^) Cantingenl 6l6 (^) mumbers Js (^) to

mauwh (^) bahind ai swny band^32 nrombaus

n a paradi. The uoo raups au Jo maudh

in he 3amL numbn e Lslwnns. what is

a mauimum nwnhan o alunns in shih

iy can manh?

Ewuise 3. dhaw hat au4 pasitive add integen tha m 64 t1, 6443 69t5 ,whuu ntagen.

Snlukon Ta hauo: Any pasiHe sdd intpe

hu um Gqtl qt 3 64 + By Eulid'> Divizion^ Lunma a -ba t et a he ay pmsie dd intg lermuma,b Sintb = passibla valuss 0,2,3,4, 0, 2 n4 thn^ o^ i>^ aNen H= 1,3 5 un a i2 cdd. 6l, 4t3 n 4t5 ea pranis Bdd paaitive integen

lse uelid's Dvian lumma o ahaw: that the

saHane any pasitt intagen is ither th aronm 3 3qt Jen samu intagen m. ialuhon: Lo shaw. ong squne pasitie intage s haum 3 30 t

Lt, poaile^ Jtagen^

be =^ a

wid's uma

a -bq 4

'b- 3

OP

.'poable Valus an 0,1 2.

a-3 , 3t 31 t

`quarúna twuns

Sose

a= (^) (x

Sas 2 a^ (31^ +)^

91+GX t

3 C3t^2 +

39t

Coheu (^) q= 3+2)

Case 3

a

= (31+2)

=91t12xt 3t

34thCwhauq 3a4zt)

A uaas^

Lan Jbe^ pLLUnted Jn^

the

05 a^ uid's^ Disian Lumma to^

show hal^

the

Lube Any ailte integg^ ia a

Ec eruie -L

Enpruss aach numban as o phadud

prima actwu

8olu 2 4o

2 40

4 35

5 5

atau oHO = 21215Xt

Soukor 2156

13 13

arta

1S6-2x2 X3 x

iiny 3825

20utor (^) S

s

3 153

tu o3825-515Y

iv 5005

Sautia

(^5) S00S

10C

1313

actou aS005-5tx UX

v 429

1313

23 23

8outio

aut o

1429

• 1}X13 23

Eind he LCM and HeF e Ling pais

intges and^ wily^

that tcM^ XHCF

paduut tws rumbus

Pars

26 and 9 7

balukerBima atkarisation 26 and 9l-

LCM 213x5x*+ X2 x

=2 3x5 Xt^ X

Go x X 966

30

2C

2

H.c.F XL.CM.= 4632

H6920-0)^32

dnd

510X32 =46920-(1i) 69 20

. H.S RHS

Hane Voiliud

i336 and 54

Selution 3 336^5

2/12 3 27

2 28

2/ 14

13

'33 2'x3xa= 2 12 x2 3xt

54= 2x

. HCF = 2x3 = 6

3

=2313x

48

LCM 2 13x 2 y 3 x

3024

No

HCF XLCM^

x

-1S144 -(1)

Bumcurtoh Numhs,

336 X54 18144-(i)

8 1

LeM XHCF - Brodurt oNo

Hana Veriia

Lind tha LCH ad HCE hu Jallauin

intwg b applying dhe puma o

Jrorigatian mathad.

12,15 and 21

(^2 12) 312X

33

',12 2x2X)

15 N 5

21 X+

HCF

2 And LeM (^) o8,9 ond 25 = 2 x^

800 'HCF -

and LCM 1800

lassusor on that HcF^ ( 306, 654)^

9, (^) ind LCMC304,5)

auion Oa^ know^ that,

LCM XHc

• Brodurt 2 numbe

LCM XHCE= 30 X .'. LCM

LCM= 31165* 22248 2233& Thus, (^) LCM (306,65) = 22,

5 Chuk uwhithn an nd shth tha digit 0

a ony^

natual (^) nwnbin. 2ouko By Eundamintal Thaaiam Aputhmeic '5 is^ not^ the

ackast

6 Th is ho sh ndual no. Jan 6 d and uaith digit o.

Q

Loploin wu^

7al? 3+13 ond

1xLx5XH *3x2X

0 ompaste hwnbus

Sxaluio lase I tXIUX13t

13 (u x1^ 4)

13 C7 +)

=13 t

Hanu, 18 is a _Compoalte nwmbU tlth mou than

2 actas

.'aXU13t13 ia s0 Csmpgslte numben.

Losa (^) I+16X5 (^) X1 3X X1t

-5(tx4 15X4 1312 XI+1)

5 ( +U).

5X+4 io09)

Hew, + js aompoaite na.with mca than

2 atas

5 XH X 3 A2X1 +5 also a Lomposte

nwmho.

3Thuna js a^ Liula^ path^ 0Dwnd^

pds

ald, asnia^ takes^

18 minutas do^ doia^

ane

SUaund

el

tha (^) dd, hilu^ Raui takes^

12 minutes

an Ama

3am (^) duppae tha

bath ataut^ at the

pom

nie- 1.

rA

hat

Js js^ jvatianal.

Ssuhion Lt, J5a^

a sakianal numben

uwhue o and b au

intagos gbto

b

Aso, a nd b orL copiunme numbes

Nous taking squASa On bath tu sdas

'. 5

'. 5b

2

a

(^2 )

b

5

2

5 diudos a

'.s diido a

Now, 0sSume tha = 5C lwhuu Cis an intug

B ) 2

2

sb-(s

sb= 25Sc

b= 5C

2

5 dvidaa b

5 divida b

.'5 (^) i» a cdmmerd (^) ack aAnd b

and b ou nt copiume numbers

whith srdadit te at hat J3 is atisna"

'sis an Juukional rumb

Haa road

hat hu

Jollir udton

A

7

Solukion t atianal no.

whe and b A opu

b

und intege and b to

'b = N2a

b anda 0w integos

and -Ji

iis alao a jutanal^ no

But, this Lodadirta thu aut that

J a bwratiana nunbo

Ou aumption was^ uran

'. is (^) iational (^) numbu

i

dsuti

Lel, J^

be n sationa (^) numba

ohu a andb Asnu

a wd dka and bt

Hee, a and b ar intege

J aJatianal nwnber

Bul =JS

5i0lss A^ atiora nunben

But this Lndradicds h ad that

J An peakonal^ numbt

Ous assumption

was uan

75 S an uatianal umher