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= 444 8, 6's 1, Find the Remainder when 77, 8-6-2 x 85 x 73 is divided by 9, (1 2 ()4 @7, 2, Find the Remainder when 273, + 375 + 478 + 657 + 597 is, = ae = divided by 25, , 5 (}) 10 (9 Ws, Find the Remainder when 1330, x 1356 x 1363 x 1368 x 1397 is, , emainder = 9 3., ler will be Zery, , divided by 13, ‘Humber So thy f)7 9 (ll ws, © the 99" term 4. Find the Remainder when 2327, O' term will be + 2372 + 2394 + 4624 + 4650, , the remainder is divided by 23, , (a) 12 (b) 14 (c) 13 (d) 10, , , , , , , , 5. Find the Remainder when 67%, + ? ; is divided by 68, s 5 fa) 67 (b) 66 (c)1 (do, elas Line ee af Remainder when 99”, : is divided by 100, 00 (a2) 99 (b) 98 (c)1 a3, 7. Find the Remainder 197130 is, divided by 196, fa)1 (0) 195 () 7 (dys, 8 Find the Remainder 6° is, divided by 215, (a) 214 () 6 (c)5 (di, 9. Find the Remainder 757575 is, divided by 37, (a1) 36 (3 (a)7, remainder is 4. 10. Find the Remainder 43197 is, inda _ divided by 7, be ioe remains ()42 41 (1 de, 62 11. Find the Remainder when 17%, hed. ny G 2 is divided by 18, fa) 17 () 16 (cy 1 (dy $4, 12. Find the Remainder, + 235 is divided by 1, 2 (1 (c) 0, 13. pad the re i ) when, he remainder of & —[7""+ 2) is diffi 6, € term. So, o wy, HS ee, . = ie er when, hainder ‘upto S| is divided b} Sus, 15, 93 2% qQ1 aa, +4, ' Find the Remainder when 2°!, (a vided by 5, 2 “1 (by) 2 (ce) 3 (a) 4, ao "6. Find the Remainder when 2#", '8 divided by 255, ; 17, @) 225 (by 128 (@) 127 (€) 64, = = 2 "(and the Remainder when 512, 5 S divided by 7, (a) 4, , mainder is 2, , (b)2 (1 ae, , n™, , a, , 18., , 19,, , 20., , 21., , 22., , 23., , 24,, , 25., , 26., , 28., , 29., , 30., , 31., , 32., , EXERCISE, ;, , The Remainder when (2)? is, divided by 3? is, , (a)8 (b)4 (c) 10, , (d) None of these, , Find the Remainder when (59)?8, is divided by 7, , (a)2 (bh) 4 (6 jd., Find the Remainder when 4177, is divided by 17, , (2 1 (6 (a4, Find the Remainder when 24°, is divided by 7, , fa)1 0) 2 (3 (a4, Find the Remainder when (51709, + 2*)is divided by 17, , a4 )5 (6, , (d) None of these, , divided by 8, (a4 2 (6 (as, Find the Remai when, 41424344454647484850 is, divided by 16 S, (a) 2 (b) 12 (8, Find th@Retainder when, 21222324 27282930 is, divided, (a. () 3 (@4, Remainder when, 3949596979899 is, livided by 16 SK yt, }3 (b) 13 (c) 11 @s, ind the Remainder when, 13233343536373839, divided by 4, 1 62 |3 @NoT, Find the Remainder when, 1234... 41digits is divided by 8, (a1 ()2 (3 (aa, Find the Remainder when, 1234....... 8ldigits is divided by 16, (a) 13 (b)8 (ce) 1 a7, Find the Remainder when 877, is divided by 17, (a)8 (b)9 (c) 13 @)7, Find the Remainder when, U+12+13+14......L100 is divided, by 5 is, (0 (1 (2 3, Find the Remainder when, UA2+13+14......1100 is divided, by 6 is, (a) 3, , , , is, , (b}) 4 (co) 2 dt, , ik, “th Yadav Readers Publication Pvt. Ltd., , 02+ 109+ 10! is, ided by 6, fa) 4 (b)6 (ce) 2 (3, Find the tay © Find the Remainder when 10!, 1234567891011121, i, , 33. Find the Remainder when, , UH2+13+(4......1450 is divided, by 12 is, , (a)2 (+) 8 ()7 9, Find the Remainder when 91", is divided by 11, , (a)2 ()9 fc) 7 (a6, Find the Renfginder when 5°, , is divi I, We 125 (d) 1, , 34,, , 35., , (a) 5, Find the Rethainder when 4010?, , is djyidedyty 7, wal © 4 ()3 (dj2, ¢ Remainder when 10!, , 36., , 37., , , , + 107+ 103+, is divided by 6, , 10! + 1 Q!001, , @4 &)6 (2 @3, 39. Find the Remainder when, 666666 ......... 134times is, divided by 13, @1 0)3 (11 @9, 40. Find the Remainder when, 555555 ......... 244times is, divided by 37, (a) 18 (b) 5 (<) 36 (do, 41,, , Find the Remainder when, , TITTATT kc, 363 times is, divided by 11, (a)0 ()7 (1 (a3, , 42. Find the Remainder when, , 888888 ..... 184 times is, divided by 37, (a)1 8 (36 (a7, , 43. Find the Remainder when, , 999999999 is divided by 13, (8 11 (5 @i2, Find the Remainder when 7%, is divided by 2400, , (a) 1 (&) 49 (¢) 343 (d)7, Find the Remainder when 31989, is divided by 7, , 2 M6 (4 as, Find the Remainder when 5412, is divided by 17, , @4 5 (3 @is, Find the Remainder when 2157, is divided by 17, , (a) 8 (b) 13 (¢) 16 (a9, Find the Remainder when 8321, is divided by 17, , (a) 13 (bk) 9 (gg, , 44., 45., 46,, 47., , 48., , (a2, , 37

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——«, , If o+ 62+ 4x +k is exactly 64, On dividing (8- 6x + 7) by (x+y, , 49, Find the Remainder when 57., © ig divided divisible by (x + 2) then value then remainder is =, (32x) is divided by 9 of kis :- a}2 (b) 12 () 0 7, (a4 0) 7 (1 (2 (a) -6 (b) -7 (c)-8 (@)-10 65. If(s- 9x24 12%- 14) is divide, 50. Find the Remainder when 58. Value of k for with (x - 1) is a by (x - 3), the remainder is ;, ie a factor of (x? - k) is :, (s2")" is divided by 7 (a) -1 o 1 is 8 (d)-8 (a) 184 (b) 56 (c) 2 , (d)1, 66. When (x*- 3x°* 2x? - Sx +, , @4 &7 2 1 59. If x'%+ 2x + kis divisible by (x, , . aoe 2, 51. Find the Remainder when + 1), then the value of k is: is divided by (x-2), they, , remainder is :, , , , , —— (a)1 (bo) -3 (ce) 2 (d)-2, (Ser) is Givided by. 60. If (x?- 5x? + 4p) is divisible by (x fa)3 Bae 2 AO, (a5 &)4 ()6 (2 +2), then the value of p is 67. If 52k 5K Ot 9 is divided, 52. Find the Remainder when (a) 7 (b) -2 (c) 3. (d)-7 (x ‘remainder is :, 888" + 222%8is divided by5. 61, If (x- a) is a factor of (x°- 3x° a (a) 135 (op -135(c) 63 (a) -63, (a0 1 (3 4 + 2a2x +b), then the value of b 68, Ifa(x! #¥1) is divided by (x + ], 53. Find the Remainder when is: in remainder is :, 22225 + S555 is divided by 7 fo 2 1 3 “ o) 0 (cy 11 (a2, 0 2 |4 MS — 69 If e+ 3x2+4x + k contain ” ae: Riedivi, 54. Find the Remainder when 2x8+ 5x2- 4x- 6 is divided h, 7 (x + 6) as a factor, the value of, ax + 1, then remainder is :, 50" is divided by 11 a aol 3, ae m4 (7 WS g © Os % a 0 a (y @) -> (b) 3, 55. The Remaidner when (20)?? is . factors of (x? + 1022 + mx¥n), divided by 17 is rer me () -3 (d) 6, fa 11 3 (6 toe yee eA ° 70, If x8 + Sx? + 10K leave, (a) ar oun ) remainder -2x when divided b, , (d) Can't determind, , 56. If (x- 2) is a factor of (+ 3qx- (b) m = 7 eu x2 +2, then the value of k is:, 2q), then the value of q is = (c) m=7,n 4 » (a) -2 (db) -1 (c)1 (2, , fa)2 b)-2 (c)-1 @I (@) "Aw, i ANSWER KEY, , 1. (b) 8. (d) 15. (c) 22. (d) 29. (a) 36. (d) 43. (b) 50. (a) 57. (c) 64. (b), 2. (a) 9. (a) 16. (b) 23. (b) 30. (b) 37. (a) 44. (c) 51. (b) 58. (b) 65. q, 3. (b) 10. (c) 17. (a) 24. (c) 31. (d) 38. (c) 45. (b) 52. (a) 59. (a) 66. (b}, 4. (b) 11. (c) 18. (a) 25. (b) 32. (a) 39. (a) 46. (a) 53. (a) 60. (a) 67. (dj), 5. (ce) 12. (a) 19. (b) 26. (c) 33. (d) 40. (b) 47. (b) 54. (d) 61. (a) 68. (b), 6. (a) 13. (a) 20. (c) 27. (c) 34. (b) 41. (b) 48. (d) 55. (a) 62. (c) 69. (cl), 7. (a) 14. (a) 21. (b) 28. (a) 35. (b) 42. (b) 49. (a) 56. (c) 63. (c) 70. (cl, , , , , , , , , , 5 +4, 1 0) Pa 3. (b), 77x 40 +4 2H 46 R=(-3)=3, 9 - + h nf aa), 5x4 1330 x 1356 x 1363 x 1368 = 1397 a 4, woe #120 13 16x-36 =-2. =R=9, ar) 3, +3 4.(b 13, 2. (2 0 +43 +7 23 t 0), A be woh 4x4x-2x3x6 _16x-36 ee . * 4, 273+ 375 + 478 + O57 + 597, 13 13 2327 + 2372 + 2394+ 4624 + 469, 25, * Avoid '-' (Negative) sign. 23, -2404+3+7-3_ 5 Normally divided 36 by 13 443424144 14, 95. 25 =5 remainder = -3. Now use '-' ie 2 23, , (Negative) sign, , I?, a, , Rakesh Yadav Readers Publication Pvt. Ltd.

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3”, , - 26. {c) 91929394959697989° 30. () 47, 3 A g?=512, -28 _ 28) (Avoid -ve sign) Last '4' digits 9 899 ,, TS ,, ms Remainder = 18) ;, Ree (6) “6 = oon ¢ oy x8? _ @12Y° x64, 7 . oe = Oe, 485 17 17, x Z 139, 21. b)- 2 =8 19 ;, i (2)_*, +1 . =, % Remainder = 11, y 319233. .000+--3839, > @yx2 22 _@) x2 27. ow 7COCOC— a x 2x64, , 7 divisibility by ¢ > The last '2', , digits divisible by 4, , 16, _G)"* x2_1x2 _2 Last '2' digits 39, 7 7 #7 39 9s Mod Cafntrs, , , , R=2 Re, , , , , , , , , , , , , , 22. (ad) O, t 1x2x-4_ -8, 5P™ 42" 2 SMe SS, Se 28. (a) 17.~«217, 88 F 1to9 = 9 Aigi R=17-8=9, 51 is divisible by 17 So (51) is ‘rom = ., divisible by 17 then remainder Remainder = Ne digits 31. (d) ules. sed [100, ‘0’, Now only divide 2”, -1 Number = 41, t ’, 1,2,3,%4..5 io fesse 41, oe _ @Y x2 Gey x2 digits os: u-1-4+.R= 1, 17 17 17 ae =9+16= 5, W.... 23 a 25 +2, _ (1)? x2 _1x2_ 2 igits = 425 s, “7 «17 «7 4 -1x0 .2.p=, R=2 Remainder -% [2=1%2 agR=2, 23. (b)1234567891011121314 R=1 +1, 8 9, (a) 1234...... 81digits 4, + divisibitity by 8 > The ta 16 [g-1x2x3--8.R =1, Three digits are sn from 1 to 9 = 9 digits, So Now last 3 digit Be Remainder digits = 81-9 = 72 “1, by 8 we get omy digits *, = 24, ois Number = 2. 36 [4=1*23x4- s R=-1, 1, 2,3, .....9/ 10 11.. 81 digits =5x4x3 120, Total Number = 9 + 36 + 45 aS eo, 1, 2,3, sseseenee 43 44.45 Ls, le, lioois |, Last '4' digits 4445 divide by 16 perfect divisible by 5, we get remainder remainder '0', 4445, "75.7 +1+2+1-100 0 0, HAHA SD rf, 25. (b) 21222324252627282930 ease _ UA2+13+ Las +L6e.. 199%, 8, Last '3' digits 930 I a 5, 93 =r 1+, Remainder = 339 125 = is2tl-1_3, 8 112 5 5, R=2 T3—> Remainder R=3, , Rakesh Yadav Readers Publication Pvt. Ltd.

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2 10 6 fO obo’ eto 10, ws : 34. ft ‘ 37 (a) z, ° eC), a % iw VW 10! e R-4, epee rRe2 ~ Cay", _(2)" 4 4 +42, tie aon ‘at, 7 > ACO ve star 10! +10? = eee “5, 2 pr sy?, Brma-Z-R-2 ze (2) x2 R-2, 1 MW, 10' +10? + PO”, 6 6, Remgindew= 0, ‘Oagill be the remainder of each, terms, will be the remainder cf 98, 22, ;, Now use -ve sign, R=-2 2 te, Actual Remainder = 11 - R=4, se Pk Rell Retna, ie ”) 36 get remainder 4, where, natural number, S'=125, , , , 6, ¢, 6, -1 B-ipa03-S,-p a6, mee, ° PY, = Me nkedeaea «, Be, Loos, ible by 5 Bniezeaeang, <o, , °c, oo, , isa, ; a 79, - 12, 5 Reg, , Jj 4“ Yate Readers Publication Pvt. Lid., , , , , 25, , Y 1x25 _ 25,, 126, , 126, , , , - we sign will be no effect, because power is even, , ay", “7, 2-8, (2) 02, “7, “, , ¢, , GY 2 (Ay n2, 7 7, ied 2, 7 7, K-2, , , , 38. (c) 10'+10° +10" Lo" + LO, +4, ‘, 10, 10’ =— R=4, 6, +4 +4, *, 10'+ 10° 10+100_&, 6, R=2, 4 th +4, ‘+ 4, 10' +10" +10! = 10 +100 +1000, 6, dead 12, 6 6, R=0, , ‘O' will be remainder of each, three terms so ‘0! will be the, remainder of 999", , ; 4, ro" 10 deg og, o 6 6, R-2, S94) 066600, 144 times, Is, , Io