LCM AND HCF. Type - I. Type - III. Type - II

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LCM AND HCF Type - I. The HCF nd LCM of two numbers re nd 9 respectively. Then the number of such pirs () 0 () () (SSC CGL Tier-I Exm. 0 Second Sitting). The product of two numbers 08 nd their HCF. The number of such pirs () () () (SSC CPO S.I. Exm 00). The HCF of two numbers nd the other two fctors of their LCM re nd. The lrger of the two numbers : () 6 () 99 () (SSC CGL Prelim Exm. 00 Ist Sitting). The product of two numbers 0. If the H.C.F. of the numbers, the greter number () 8 () 0 () 0 (SSC CGL Prelim Exm. /0/00 (IInd Sitting). The HCF of two numbers nd their LCM 00. If one of the number 60, the other : () 0 () 6 () 00 (SSC CGL Prelim Exm. 08/0/00 (Ist Sitting) Type - II. Three bells ring simultneously t.m. They ring t regulr intervls of 0 minutes, 0 minutes, 0 minutes respectively. The time when ll the three ring together next () p.m () p.m. p.m ().0 p.m (SSC CGL Tier-I Exm. 0 (Ist Sitting). The lest number, which perfect squre nd divible by ech of the numbers 6, 0 nd, () 600 () 600 600 () 00 (SSC Section Officer (Commercil Audit) Exm. 00 Second Sitting). The lest multiple of, which on dividing by,, 6, nd 8 leves reminder in ech cse : () 0 () 8 () 80 (SSC CGL Exm./0/00). Find the lrgest number of four digits such tht on dividing by, 8, nd the reminders re,, nd 0 respecitvely. () 6 () 6 6 () 66 (SSC CGL Exm./0/00 (Middle Zone). Three men step off together from the sme spot. Their steps mesure 6 cm, 0 cm nd cm respectively. The minimum dtnce ech should cover so tht ll cn cover the dtnce in complete steps () 960 cm () 960 cm 690 cm () 690 cm (SSC Tier-II Exm. 0) Type - III. The mximum number of students mong whom 00 pens nd 90 pencils cn be dtributed in such wy tht ech student gets sme number of pens nd sme number of pencils, : () 9 () 90 00 () 9 (SSC CGL Prelim Exm. 0/0/999 (Ist Sitting). HCF of, nd 6

() 8 0 0 () 0 () 0 (SSC Grdute Level Tier-II Exm. 6/09/0). The gretest number, which when divided by 989 nd leve reminders nd respectively : () 8 () 6 () (SSC CGL Prelim Exm. /0/00 (IInd Sitting). Wht the gretest number tht will divide 0 nd 0 leving reminders nd respectively? () 9 () 6 () (SSC CGL Prelim Exm. //00 (IInd Sitting). Find the gretest number which will exctly divide 00 nd 0. () 0 () 0 6 () 0 (SSC GCL Tier-II Exm./09/0) Type - IV. The LCM nd HCF of the numbers 8 nd re in the rtio : () 6 : () : : () : (SSC CGL Prelim Exm. /0/000 (IInd Sitting). Three numbers re in the rtio : : nd their HCF. The LCM of the numbers () () 9 96 () (SSC CGL Prelim Exm. 0/0/00 (IInd Sitting). The rtio of two numbers : nd their HCF 8. Then their LCM () 0 () 0 0 () 60 (SSC CGL LDC & DEO Exm. 0//0 (IInd Sitting - North Zone) Type - V. The LCM of two numbers times their HCF. The sum of the HCF nd the LCM 0. If one of the number 9, then the other number () () 8 () 8 (SSC CGL Prelim Exm. /0/008 (IInd Sitting). The sum of two numbers 6 nd their HCF nd LCM re nd 0 respectively. The sum of the reciprocls of two numbers () () () (SSC GCL Tier-I Exm. 6/0/00). The sum of the H.C.F. nd L.C.M of two number 680 nd the L.C.M. 8 times the H.C.F. If one of the numbers 6, the other : () 8 () 8 () 96 (SSC GCL Prelim Exm. //00 (Ist Sitting)

Type - I. Let the numbers be x nd y where x nd y re prime to ech other. LCM = xy xy = 9 xy Possible pirs = (, ) nd (, ). Here, HCF = Let the numbers be x nd y where x nd y re Prime to ech other. Now, x y = 08 08 xy The possible pirs re : (, ), (, ), (, 6). () Let the numbers be x nd y where x nd y re co-prime. LCM = xy As given,. () xy x, y The lrger number = y Product both number LCM HCF 0. () st number IInd number = HCF LCM 00 so II nd number 60 Type - II. () LCM of 0, 0 nd 0 minutes = 0 minutes Hence, the bells will toll together gin fter hours i.e. t p.m. EXPLANATIONS. () The smllest number divible by 6, 0 nd = LCM of 6, 0 nd 6, 0, 8, 0,,, 6,, LCM = Required complete squre number 600. LCM of,, 6, 8 = 80 so the number will be (80k + ) divible by K = so the required number 80 + = 0 + =. () Solve nd find out the LCM seprtely 8 LCM of these 89 0 The lrgest number of four digits = 9999 divided 0)9999( 60 9 reminder so the required number = 9999 9 = 6 8 0

. LCM of 6, 0, sme method s upon question = 690 m Type - III. () mximum number of students = HCF of 00 nd 90 = 9. () HCF of,, 6 HCF of,.6 LCM of,, 0. 989 hs reminder so 989 = 98 = 0 Required HCF of 98, 0 HCF = 98, 0 98)0( 98 6)98( 6 )6( )(. () The number = subtrct reminder 0 0 HCF 0 0)( 0 9)0(6 9 So the number = 9. () HCF of 00 nd 0 00)0( 00 0)00( 0 80)0( 0 0)80( 80 Required number = 0 Type - IV. () LCM of 8 nd 8 8 Now HCF of 8 nd = 8 rtio 6 :. () We consider the numbers x, x nd x respectively HCF = x = Numbers re = = 6 8 Now LCM of, 6, 8. () Number be x nd x HCF = x = 8 Number = nd 0 LCM = 0 6 0 8 0 8 60 Type - V. () HCF of number = H LCM = H in the question given H + H = 0 H = 0 H 0 LCM

Now I st number II nd number = HCF LCM = 9 II nd number II nd number 9. Let the number be x nd y x y 6 x y...(i) xy 0...(ii) dividing eq (i) by (ii) x y xy xy 0 y x. () HCF be h LCM be l l = 8 h nd l + h = 680 8 h + h = 680 680 h 8 8 l = 680 8 = 6 Now the other number = 6 8 96 6

SIMPLIFICATION Type - I. The vlue of 9 (). If x () () SSC CGL Prelim Exm. /0/00 (Ist Sitting). Simplify 0. () () 0 () SSC CGL Prelim Exm. /0/00 (IInd Sitting) 9 equl to () (). (). SSC CGL Prelim Exm. 08/0/00 (Ist Sitting) then, the vlue of x () () () 6 SSC CGL Prelim Exm. /0/00 (IInd Sitting). simplify 9 () () 9 9 8 () SSC CGL Prelim Exm. /0/00 (Middle Zone) 6 6. The vlue of. () () () 0 (SSC GCL Tier-I Exm. 6/0/0) Type - II equl to : 9 0 () () 6 () (SSC CGL Prelim Exm. 00 (IInd Sitting)

.. simplified to () 0 () 0 () (SSC CGL Prelim Exm. 00 (IInd Sitting)..0.0....0.....0....0.0. simplified to () 0 () () (SSC CPO S.I. Exm 00). If * represents number, then the vlue of * in. 9 : * () () 6 () (SSC CGL Prelim Exm. 00 (Ist Sitting)...9 9... equl to () () () (SSC CGL Prelim Exm. 00 (IInd Sitting) 6. The lest frction to be subtrcted from the expression of 6 to mke it n integer. 0 () (). If 0.0 0.x 0.0 6 0. y,find the vlue of x y () 0 (SSC CPO S.I. Exm 009) () 0.00096 () 0.0096 0.096 () 0.96 (SSC CPO S.I. Exm 009) 8. Simplify : 0.0 0.0 0.0 0.96 0.0 0.0.096 0.96 () 0.906 ().0009.000 () (SSC CGL Tier-I Exm. 0 (Ist Sitting) 9. The vlue of 0... 0.008. 0.6 0.0 () 0 ().99.08 () (SSC CGL Tier-I Exm. 0 (IInd Sitting) b b () b () b () 0 (SSC (Grdute Level Tier-I Exm 0 Second Sitting) 6 () () () 9 (SSC CHSL LDC & DEO Exm 0 Second Sitting

. Find the sum of n n n... n n () n () n n (SSC (Grdute Level Tier-II Exm 0) n (). The vlue of 0 9 () () 6 8 () 8 99 (SSC CGL Tier-I Exm. 0 (Ist Sitting). If x,then x () () () (SSC CGL Tier-I Exm. 0 (Ist Sitting). The vlue of 8 8 () 9 () 60 () (SSC CAPFs SI, CISF ASI & Delhi Police SI Exm 0 ) 6. The simplified vlue of of 6 0 8 () 6 () () (SSC CAPFs SI, CISF ASI & Delhi Police SI Exm 0 Second Sitting 8

. () Type - I 9 9 EXPLANATIONS. () 9 solve from to 9 or 9 now 9 or 9 9 9 9 9 = 9 9. x 8 x 8 8 or x 8 0. (). () 0 or 8 8 8 tke the LCM & solve 0 9 9 9 9

9 9 9 0 9 or 9 9 6. () or or 0. () Expression Type - II 0 0 0 6 0. () Expression 0 0 8 0 60 0 0 80 0 8 60 0 90 90 0 0 0. () Let (..0) =. () (.0.) = b nd (..) = c b c 0 Expression b c bc c b b c bc bc bc [If + b + c = 0, + b +c = bc] 9 * 9 * 8 * *. () Expression...9 9....9 9... 9 9 6. () Expression 6 06 0 0

6 06 0 9 8 60 9 66 0 0 Required nswer. () 0.0 0.x 0.0 0. y On squring both sides, 0.0 0.x 0.0 0. y x 0.0 0. 0.0096 y 8. () Let 0.0 = nd, 0.96 = b b Expression b b b b b b b b 0.0 0.96 9. () Expression. 0... 0. 0. Let. = nd 0. = b b Expression b b 0. b b b b b b. 0. b b b b b b b b b. () Expression 9 6 9 9 6 9 9 9 9 6. () n n n n... n

n n... n n n n... n n n n n n n n n n n. () Expression 6 0 9 6 9 0 6 9 0 6 9. () x[ { ( )}] [ { ( )}] x 8 x x x 6 6 x. () Expression 8 8 6 8 6 60 60 0 60 0 6. () Expression of 6 0 8 0 6 0 6

. Squre & Cube Root TYPE - I 0.0 0.0 The vlue of : 0.0 0.0 () 0.0 () 0.00 0. () 0.0 (SSC CGL Prelim Exm. 999 (IInd Sitting). The vlue of 6. 6. 6. 0.6 6. 6. () 0. (). 0 () 00 (SSC CGL Prelim Exm. 00) () 0 ().96 () 0.0 (SSC CGL Prelim Exm. 00 (Ist Sitting). The smllest - digit number, which perfect squre, () 009 () 06 0 () 0 (SSC CPO Sub-Inspector Exm 00) 8. The lrgest number of five digits, which perfect squre () 99999 () 9996 9986 () 996 (SSC CGL Prelim Exm. 008 (Ist Sitting) 9. Wht number must be dded to the expression 6 to mke it perfect squre?. The vlue of 9 9 9.. () () 9 () (SSC CGL Prelim Exm. 00 (IInd Sitting) 0.009 0.06 0.06 0.08 0.00 0.0008 0.000 equl to () () 6 8 () 9 (SSC CGL Tier-I Exm. 00 (IInd Sitting) () 0.0000 equl to : 0.00009 () () (SSC CGL Prelim Exm. 00 (Ist Sitting) 6. 0.98 0.0 0.98 0.0? () 9 () () 6 (SSC CGL Tier-I Exm. 0) 0. Three fifth of the squre of certin number 6.. Wht the number? () 0. ().69. () (SSC CGL Prelim Exm. 00 (Ist Sitting). If 6 nd b 89, then the vlue of b b () / () () (SSC CGL Tier-II Exm. 0). The vlue of 6 0 0. () 0.00 () 0.000 0. () 0.0 (SSC CAPFs SI, CISF ASI & Delhi Police SI Exm 0)

. The squre root of () () None of these (). The vlue of (SSC CGL Tier-II Exm. 0) () () () (SSC CGL Tier-II Exm. 0) TYPE - IV. The sum of the cubes of the numbers, - nd - equl to () 690 () 960 () 0 (SSC CPO Sub-Inspector Exm 00). 6 equl to () () () (SSC (South Zone) Investigtor Exm 00). The sum of the squres of numbers 6 nd the squre root of one of them. The cube of the other number () () () (SSC CGL Prelim Exm. 00 (IInd Sitting). equl to () 9 () () (SSC CGL Tier-I Exm. 0 (IInd Sitting). The lest number, tht must be dded to 0 so s to obtin perfect cube, () () 8 () (SSC SAS Exm. 00) 6. The lest possible vlue of A for which 90 A perfect cube () 00 () 00 00 () 600 (SSC CPO Sub-Inspector Exm 00). If the squre root of x the cube root of y. then the reltion between x nd y () x y () x y x y () 6 x y (SSC FCI Asstnt Grde-III Exm 0) 8. The sum of the cubes of two numbers 9. The sum of the numbers. Then the difference of the two numbers () () 6 () 8 (SSC CGL Tier-II Exm. 0) 9. If the cube root of 90, then the vlue of 9.0 0.090 0.000090 () 0. ().. (). (SSC CGL Tier-I Exm. 0 (IInd Sitting)

EXPLANATIONS 0.0 0.0. () 0.0 0.0 [Using b = ( + b) ( b)] 0.0 0.00.0 0.0 0.0 0.0 0.0 0.0 0.0 8 0 9 9 8 0 6. () 0.98 0.0 0.98 0.0. 6. 6. 6. 0.6 6. 6. 0.98 0.98 0.0 0.0 0.98 0.0 0 0.6 06. 0 6. 0.6 6. 6. 00 0. () 9 9 9 6 9. () Expression. () 0.009 0.06 0.06 0.08 0.00 0.0008 0.000 9668 8 6 0.0000 0.00009.000. The smllest -digit number = 000 The smllest digit perfect squre number = 0 = 0 8. Lrgest -digit number = 99999 Now, 99999 6 9 6 99 6 66 99 6 6 6 Required number = 99999 = 9986 9. () b b b 6 9 Hence, on dding, expression will be perfect squre.

0. Let the number be x According to the question of x 6. 6. x x 0. x 0... () 6 nd b 89 6 8 nd b 89 b b 8 8 9. () Expression 6 0 0. 0. 0 0 0 6 6 0.0000 8 0 0.. Type - II. () Here, = 0 We know tht b c bc, if b c 0 690. () Expression 6 6 6 6. First number Let the second number be x. x 6 x 6 x Cube of =. () Expression 6

6 6 6. () = 8 Required number = 8 0 = 8 6. () The number = 90 A A The lest vlue of A for which the given number perfect cube 00. () x y x y 6 x y x y 6 8. () Let the numbers be nd b b 9 b b b b b 9 b 9 9 9b 9 9 9b 9b 0 b 6 b b b b 6 69 b 9. () 90 9.0 0.090 0.000090. 0. 0.0.

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