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UNIT DIGIT, , , , , , Last Digit of number is, Digit, 1234, , called Unit, , unit digit, In This no. 4 is unit digit., The unit digit of, ) Resultant value depends u, The unit digits of, participating numbers., Ex.l: 23 + 34446478 = 181, unit, digit of 181., +. unit digit = 1, It is clear that the unit digit of, the Resultant value 181 de, pends upon the unit digits 3,, 4,6,8, , 3+4+6+8=21, So, units digit = 1, Ex.2: What is the unit digit of ?, 34 31 x 37 x 36 x 46 x 89, sol 31 x 37 x 36 x 46 x 89, 12 Unit digit = 1, 7, 6,6,9, , the, pon, all, , 58 multiply the unit digits = 1x7, x6x6x9Q, 80> 1x7=7, > 7x6=42, 61= 2x6=12 ., 407 2x9=18 Cc, , ett,, , unit digit = 8 AAD, 68223 What is the unit digit 3 \) ", 31 « 33 x 37 x 39 x/4g, , TAsg 31% 33x 374 30%, , , , , , multiply, =1*3 x 73@g\x, unit digit = 7, , = 37°84: What is the Unit digit of ?, , 2 91 * 93 x 95 x 96 x 97 x 98, 56251. multiply the unit digit, 626Ex.5;, , 1x3x5x6x7x8=0, Find the unit digit of 135 « 136, 650301,, , ~170, , The unit digits = 5, 6, 0, ultiply the units digit, “Sx6xQ, , * unit digit = 0, , , , , , Ex.6: Find the unit digit at the, Product of all the o, , dd prime, numbers,, , sol. The prime numbers are 3, 5, 7,, 11, 13, LT pees etc., , Now we know that if 5 is multiplied by any odd number it always gives the last digit 5. So, the required unit digit will ne, 5!, , Ex.7: Find the unit di, , git of 584 x 22, x 547 x 613 *, , Sol. The unit digits = 4, 8, aM:, , multiplying the unit digits, , ree Ra, , = unit digit = 2 ay, , Ex.8: Find the unit digitof the prod, uct of all the even numbers, , The evernn yr, , 2, 4, 6, ao, se ete., , Now we kndw that if 0" is multi, Plies number it always, , an ae digit 0. so the re, quire@arnit digit will be 0., Ex.9: Findthe unit digit 4!, Sol. \4! ='4 x 32% 1=24, : nit digit = 4, 103 Find the unit digit 5!, “S!=5*4x3x2*1= 120, unit digit = 0, Factorial 5 and more than 5, express gives unit digit 0., , Unit digit when 'N' is Raised toa Power, unit digit of 0, 1, 5 and 6 has, any power (odd or even) no, change, , Ex.11:(3765)"”, unit digit = (5)*7= 5, , Ex.12: (6736)°5°7, unit digit = (6)9?5°7= 6, , Ex.13; (32541), unit digit = (1)5= 1, , Sol., , , , , , , , * 4and9, 4 2, even odd even odd, power/ \power power/ \ power, i unit, oat 6 4 digit >I 9, , **kesh Yadav Readers Publication Pvt. Ltd., , \, , Ex.14: Find the unipplace (67354) !297, Sol. (67354)23"%), , wit fan = (4) odd power, , Sol. (3259)'hs, i, A it digit = 1, , . 16) Find the unit place (6734)?”?, Bohy © (6734)312, \. unit digit = (4)312 = (4yeren, unit digit = 6, Rule of (2, 3, 7 and 8) >, unit digit when 'N’ is raised to, a power, , If the value of the power is, , Power, Sn, , unit 1 2 3, digit} or or or, J | 4n+1]/4n+2 |an+3, 2 2 4 8 6, 3 3 9 7, 7 r 9 3, 8 4 2, here n + Natural No., If those number which unit, digit 2,3, 7 ands. 5, all unit digit have cyclicity 4, Ex.18: Find the unit place 3°5, , , , , , , , , , , , , , , , , , , , , |, , , , , , , , , , 1, 1, 6, , , , , , Sol. 3%5= 322 x 33, Break the power form of 4n, (34)8 * 38 = (ay x (...7), , unit place = 1x 7=7, Ex.19: Find the unit, , place (127)39, Sol. (127)°, unit place = (7)39, = (P= (7p = (Pe ers, = (0.1) ® (1.3), , unit place = 1x 3=3, Ex.20: Find the ‘unit place (678)s02, , Sol. (678)s02, unit digit = (8)5e2, = (8) x (8)?, = (8°) x (g)2, = (...6) * (0.4), , unit digit= 6 x 4=24 =4, , I

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Ex.21: Find the unit place (327), , Sol., , (327)e°, , unit digit = (7), 640 is multiple of 4, then = (74!, , unit digit = (1)'= 1, , Ex.22: Find the unit digit of (2137)', , Sol,, , (2137), , unit digit = (7/8, = (7)2x 71, = (74) x 7!, = (1) %7, , unit digit= 1x 7=7, , Ex.23: Find the unit digit of (13)?0?, , Sol., , Sol., , , , (13922, unit digit = (3), = 3200 x 39, = (34) * 3?, = (42.1) * 27, =1%*27=27, unit digit = 7, Ex.24: Find the unit digit of (22), (22), unit digit = (2)?, = (2)9x 2°= (245% 8, = (....6)5* 8, , unit digit = 6 x 8= 48=8, , Bx.25: Find the unit digit of (37)!, , Sol., , (37)!, , unit digit = (7)!, = (7)! x 7!, = (71) «7!, = (17, , unit digit = 1* 7=7, , Ex.26: Find the unit place, , Sol., , 3, Note:- unit digit = even x 5: ='0', Ex.27: Find the unit plac, , Find the unit digit of 584 x 389, = 476 x 786, , @7 @)3 (4, Find the unit digit of, 641 x 673 x 677 x 679 « 6814, , (ae, , (a)9 (3 ()6 (d)7, Find the unit digit of, (5627)!© x (671)? AS, @7 09 |X +, Find the unit digit Ai 7, (3625) x naeeN ”, (a) 6 (b) 3, apt. %y) fd) 0, Find the uni, , (3694)!793 x & (841)!, (a) 5 ) 4 (do, Find the git of, , (7% - 3°) 9, , fay 7 ()3 (c) 4 (dO, Find the unit place of, , (17) + (11)! — (7), (a2)O0 (b)1 (2 (d)7, , Find the unit digit of, 36 « 47 x 6 x 7* x BK 98, , (a)6 (b)9 (c) 0 (d) 2, Find the unit digit of 111! (factorial 111)., , (gO (b)1 ()5 (A) 3, , v, , 10.“the., , , , , , , 12., , 13., , 14,, , 15., , 16., , “Val Oo (9, 1.dFind the units digit of the, , (23)?! x (24)22% (26)? « (27)? x, (25)5, , (23)?! x (24)22 x (26)?9 x (27)™ x, (25)°5, , unit digit = (3)?! » (4)? x (6)9 x, (7/24 = (5), , Break the power multiple of 4, , 3° x 3! x 4% x 6” x (74° x 5%, , rte ad \, even same same, power digit digit, , 4, , =3x6x6x1x5 “|, , unit digit = 0 eh, , , , (a167)", ‘unit digit fc, , wy, Ag Md, , , , aaeuigit of the number obtained t by multiplying the numbers 81 x 82 x 83 x 84 x 85 x 86, x 87 x 88 x 89 will be, (c) 7, , , , (d) 2, , expression 25°! + 365% + 2285, (a) 4 (b)3 (6 (d)5, Find the units digit of the, expression 5575 + 735810 + 22859, (a) 4 (0 ()6 (5, Find the units digit of the expression, 11! + 12? + 139 + 14% + 155 + 16°,, (a)1 (b)9 (d)7 (dao, Find the last digit of the number, D3 25 89 40 O08,, , (a) 0 (b)1 (d)2 (5, Unit digit in (264)'°7+(264)' is;, , (a)0 (b)4 ()6 (8, Unit digit ((251)+(21)%(106)'°+(705)"5-(16)'+259] is, fa 1 (b)4 (5 (6, , . The unit digit in the expansion of, , (2137) is, , (a1 (bF)3 (7 OD, , Rakesh Yadav Readers Publication Pvt. Ltd., , Ex.28: Find the unit place of, , Sol., , , , , , , 19,, , 20., , 25., , - what is the number of unit pl, , (er + ay + yrs (ry 26, , | |, , , , | 27, , , , , , , , , , , , , , , , , , , same G22, Same, i, =5 +6 + 6+1 28., , unit digit = 18 = 8, , 29., 1x ¢ |, , an 7, eb, , 30., , , , , , ~) | 31., “Ral, “(ax4y® (2 3, a7 ap 4 32., 355x455 git, =o + 4133., = 3x 455 4.972, unit digt = (.....1) * (......4)4, =4+6=10, 434., unit digit = 0, 435., . The digit in unit’s place of f, product (2153)'® is :, (1 (3 (7 (A), The digit in unit’s place of, product, (2464)!795 x (615)3!7 x(131)# 1., f@O (2 (3 S| %, What will be the unit digit inf 4°, product of 71° ? 5., , @5 ()7 (9 (dd?, , in (329), , (a) 1 (b) 7 (ce) 9 (d), , + unit digit of the number (24_, is: |, 4 m6 (6 We, , . The unit digit in the prod, (122)"79 is:, , (@2 (b)4 (je (@%, , - The unit digit in the sum of (127°, + (124)3is :, , 5 m4 «2 @®, The last digit of (1001)°* +(2, Mo m3 4 @ ‘, , yes, , 1

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SSEa==—————, , , , , , , , , , , , a 96. Find the unit digit in the product: 36, Find the last digit of, (4387) x (621). 22084 Bgg222, (a) 1 oo) 2 ee (a) 7 (a) 2 (bF)6 (JO (as, : it digit of t < i, BP oan BSE TIM ig 87a Bind the last digit of ase, . (96 (5 () 4 (do 4 (b)8 (6 (a2, 28. The unit's digit in the product 77" 38. Find the last digit of the expres- 45., x 68 x 3S is sion:, @1 ()2 (3 (da 17+ 22+ 32+ 424... + 1002,, 29, The last digit of 3°° is ()0 ()4 (6. (8, a1 (013 (7 (a) 9 39. Find the unit digit of 1! + 22+.33 46., 30. The digit in unit’s place of the + see 1010., f :, number (1570)? = (1571)? + (a)9 (b)7 () 0. (d) N.O.T., (1572)? + (1573)? is : 4 f yoatbes, (4 (1 (2 3s 0. Find the unit digit of, 4, 31. The unit digit in 3 x 38 x 537 x Sor 68°" + 2413 x 576+ 1234+ 47., 1256 is ., 4 ()2 (6 (as (a) 4 )7 (c)O (dd) 8, 32. The unit digit in the product 41. The unit digit of the expression i, (2467)'® x (341)7is. 1x2x3x4x5x6x7x8x9x10 |, @1 3 ()7 (9 =... o7> > 7 4, 33. The unit digit in the product 4 ., (6732)! x (6733) x (6734)""" (7 (b)9 (c) 8 (d) c(N, (6736}'76 42. Find the unit digit of the expres- 49i 9235! ey, + @1 03 |4 @s Saggeer t AEgOnee, 34. Find the unit digit of the prod- . &, uct of all the prime number be- JS ()9 (©, tween 1 and 99999 (d) None of these, {a9 (\)7 (c)0 (d)NOT 43. The last digit ofjtHle following ex- 50., _« 35. Find the unit digit of the prod- pression A My! + (21)? + (31)? ;, 4 uct of all the elements cf the set + (418+ 2... BRN?, which ¢ .Jisis all the prime (a) 4 (B'S) (c) 6 (a) 7, numbers greater than2butless 44. Theé‘Tast § digits of the following, th than 222. onkaenigt will be, @4 )5 (Jo ToT, (1H (2 + (38+ (aye + (Sy +, : Y, 7 ’j | ANSWER KEY, is 1. @) | 6. (c) 11. (b) 16. (b) fj 21. (a) 8 26. (a) 3 31. (d), an) 7. (b) 12. (c) 17. (@) fy 22. id) (| 27. (a) © 32. (c) ®, n 2 @) 8. [a) 13. (b) jg) 18. (c) “@ 23. (a) 28. (d) 33. (c), 4 (a) 9. (a) 14. (a) 19. (a) 24. (a) FH 29. (a) 34. (c), 5. (@) 10. (a) 15. (a) 20. (b) wf 25. (b) & 30. (a) 35. (b), Ls, ., lac NV, SOLUTION, , , , , , (d) 584 x 389 x'476 x 786, ; unit digit 4, o.6, 6 3., jul Multiplying the unit digit, =4x9x6x6, unit digit = 6, (a) 641 « 673 x 677 x 679 681, unit digit = 1,3)7,9; 1, i Multiply the unit digit *, , =1x3%7xQ9x1, =21x9=189, , 2 Rakesh Yadav Readers Publication Pvt. Ltd., , unit digit = 9, (a)(5627)'* « (671), unit digit (7)!5 « (1)2, = (7)? « 7x 1, , = (74) 7x 1, , & (..01)8* 7, , unit digit = 1* 7=7, (d) (3625) x (4268), unit digit (5)° « (8), = 5 x (8)**x 8!, , (101)5 + (100!)4+ (1000!)9 +, (10000!)? + (100000!), , (a) 45939 (b) 00929, , (c) 20929, , (d) Can't determined, , The unit digit of the following, expression (1!) + (2!)% + (31)97 +, (99!) is:, , (a) 1 (b)3 () 7 (A) 6, , The unit digit of (12345k)’is 6., The value ®f, , fa) 8 Oy c) 2, , (d) all, , The unit digit of the expression, , (UML A (21)? + (37 + +(100!)', (b) 1 ()2 (d)7, , Me othe ee, , last digit of the expressioin, ‘49° x QI% is ;, , (4 (b)6 ()9 (dl, The last digit of the expression, 4492+ 494954 454964, 4” + 9100 is:, , (a2)0 (})3 (5, , (d) None of these, , The unit digit of 9° x 34° 4s* x, , 5° * 6” x 78" is:, , (a) 0 (oe) 5, (c) Can't be determined, (d) None of these, , 36. (c) Al. (c) 46. (d), 37. (c) 42. (b) 47. (a), 38. (a) 43. (d) 48. (b), 39. (b) 44. (b) 49. (a), 40. [a) 45. (c) 50. (a), , = 5 x (B41 x gt, 5x (6) x g, , unit digit = 5 x 6x8=240=0, (d)(3694)17%% (615)9"7 x (841), unit digit (4)!79 x (5)37 x (1)91, oud power = g, , 5°=5, , 4x5x1=20, , Hence, unit digit = 0, , 8, 3

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6. (c)795 = 388, = 792 x TI BS x Bz, = (748 x 343 - (34) x 9, = (004 1X I= (TMD, unit digit = (.....3)- (0... 9), =13-9=4, , 7. (b)(17)!99 + (1.1) 1999 — (7) 19%, tanit digit = (7) + (1)! — (7), , *» (7)9% — (7)'°9 gives = 0, , Then, unit digit = 1, , 8. (a)Unit digit = 3° x 47 x 6 x 7* x, 8? x 95, , The unit digit of 3¢ = 3¢ x 3?=9, The unit digit of 47 = 4, , The unit digit of 6° =, , The unit digit of 7* =, , The unit digit of 8? = a, , The unit digit of 95 = 9 x 91=9, , multiply the unit digits = 9 x 4, x6x1xX4x9, unit digit = 6, , 9: falll!i=1x2x3x4x5*x...., *100* 111, , Since there is product of 5 and, 2 hence it will give zero as the, unit digit., , Hence the unit digit of 111! is 0 (zero)., , 10. (a) 81 = 82 x 83 x 84 x 85 x 86, * 87 x 88 x 89, , Unit digits = 1, 2, 3, 4,, Product of unit digits, =1*x2x3x.., , , , Then, we gets unit digit =, , 11. (b) 25°25! + 3658 + 20852, unit digit = (5) a, , unit digit = (..1%45)+4(..Yg)+ (2ftd, , , , (Se (op) PR « 2, =5+6 pe 2, Sum of unit digit= 5+6+6 «2, , =5+6+12=23, Hence, unit digit =3, , 12. (c) 55725 + 735810 + 22859, unit digit = (5)’* + (3)? + (2)8%*, = (...5) + (39) x 32+ (2429 21, = 5 + (1)!952 x 9 + (16)2!9 x 2!, Sum of unit digit= 5+ 1*x9+6, *2=5+9+12=26, unit digit = 6, , 14,, , 15., , (b) 11!) + 12%+ 139+ 14° + 155+ 16°, unit digit = (1)! + (2)? + (3)? +, (a)' + (5)5 + (61°, , Sum ofunit digit=1+4+7+6, +5+6=29, , unit digit = 9, , (a) 19+ 29+ 33+ 49.... +999, Sum of cube of natural no., , nnttyy (sea), “eye, , 2, . (eo) = (99 x 50)?, = (4850)?, Unit digit = 0, (a) (264)' + (264)!, unit digit, Y54454, #5 16+ AC, , NN, , O, , Rule:When 4 has wer then unit, , , , digit is: “ Xs, When 4 as’ even power, then unit, , t ig6 6h i, ee aon _, , cage + a, iu wv, , (even power), unit digit6 = +, , (odd power), 4=1040, , Alternate :, , 16, , => (264)!2 + (264)!, , = (264)! (1 + 264), , => (264)!°? x 265, Multiplication of 5&2=0, Hence, unit digit is 0., , (b) [(251)% + (21)? — (106)1° +, (705)* - (16)* + (259)|, , unit place of 1,5 and 6 will, remain same, , = [(1)*+(1)?7{6)'+(5)5-(6)*+9], = [1+1-6+5-6+9], , => 16-12=4, , Hence, unit digit = 4, , Rakesh Yadav Readers Publication Pvt. Ltd., , 17. (d) (2137), = (7)’* will give unit digit, unit digit, 7} =>7 754 divide by 4-25, ? >9 4, 73 4 So, remainder j, 2, om 2401 > 1 nes, 7>= 16807-> 7, & will, , repeat, , Unit Plage fo, 18. e =a, gle7, init digit, a3 33, a3 +9, 2 +27 +7, Qe 381 41, This cycle will continue, => divide the power of 3 by 4, , 167 i 3, «+= remainder is 3, , S=7, Unit digit = 7, 19. (a) (2464)'79 x (615)! x (131), 44434 34, 47 = 16 +6, 5 64 34, So odd power of 4 will have 4 as u, digit and even power will have 6 |, , unit digit 5 and 1 have same unit di, its respectively, , , , (2464)'™x615)27 *(131)™, odd power, unit digit >4 x5 x =2, => 20 = 0 unit digit, 20. (b) 715, > TAT37, > P4959, => 7 +34333, = 7 +240141, Divide power of 7 by 4, 105, , 4 ~ remainder = 1 = 7! is left, unit digit = 7, , 21. (a) (32978, , => __ Ifpower of 9 is odd, then unit di, , number be 9. If power is even th, ‘unit digit number be 1., Hence, unit digit = 1, , I

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(d) (22)8, 22. (a) ¢ Poult unit digit, 2° 8, 4 * Cycle, 8 8 completes, 6, 2, , 16, 32, , So divide power of 22 by 4, , PNW, , 4, 2=8, unit digit = 8, , 23. (a) (122)'"8, Unit digit, , Ae Se, , 3%, , remainder 3, , 2°> 32 + 9], QS = QI 2E49x9 = 16 43x9, = 6%x2 = 6x2 =12, unit digit = 2, 24 (4) (zap? (124), + +, a a, When 4 has odd power then unit, digit is 4 when 4 has even power, then unit digit is 6, Ps. 4G, 16 +6, 7 geen y euervay |, , Po, , 5 2565 6, , , , , , 4st? 4 373, , tL +, , 6 + 4 oO, last (unit) digit =, , 25. (b), , tL, Unit digit . 1290 + 1002 Sy, Unit digit will be 1 in4 ol, Tespective of power ww, = 1+ 1002 = 1003 Me %, , (1001)2% + 1002, , +6 on igith= 3), , , , , 7° 24017 1, (4387) x (621)7, t, (77 x (1), 1, (7)tro181 x ], , L, p (Nron7 1, , t, (x7 1, Unit digit = 7, , , , 27, (d)5 always gives unit digit 5, and 6 always gives unit digit 6, unit digit, , 25%" + 36" 4 79H, , 574 gi 4 BH, . ‘ ‘ V54 <0, unit digt>5 + 6 + 3?, =5+6+9=20=0, Hence, unit digit =0, 28. (d) 7? x 68 x 36s, L 4 +, unit place 7? e 3!, , +, unit digit > 3 x 6 x 3 = 4, =4, (a) 3%:, , 29., , , , 4, , 1 AO, 2 AA Y Cycle, oe, , e, , 3, 3, 3, £3" aY >, Hexd@amnit digit of 3” of completipg all cycle = 1, , (1870)? + (15717 + (is72f + (1573)?, + +, , ’ t +, , ube digits g 2, Bere ug?, + t + +, Qa Tat Fg Ng cay, unit digit = 4, , 31. (da), *® 53] «125©, , * 220 +®, Note:- Always multiply only unit digit, of first no. to second and Product's, unit digit no. with 3rd no. Again Pproduct of last's unit digit to fourth and so, on., Hence, unit digit = 8, , 32. (c) (2467) x (341)”, ("aye, , [153/4= remainder = q, , ss Dx1=7, , Result —_ Unit digit, Pp . 7 7, P = 49 9, Po o= 343 3, 7 = 2401 1, , Hence, unit digit = 7, 33. (c) (6732)! x (6733)!72 x (6734)!, , , , x (6736)'76, unit digit = (2)!79 x 43}!72 x (4)!74 x (6)176, = (24)"2 x t)°9 x (4)!74 x (6)178, , = (...68% 4 x1) x (...6) * (...6), Multip ion of unit digit, = 6% 4x 126 x 6 = 864, Hence, unit digit = 4, 34. 4c) She set of prime number S, 23,5, 7,11, 13, ..... }, C@there is one 5 and one 2 which, 10 after multiplying mutually,, iffheans the unit digit will be zero., Hence, unit digit = 0, 35. (b) The set of required prime, number = The set of reqired, prime number, = (3,5, 7,11, ....), Since there is no any even number is, the set so when 5 will multiply with, any odd number, it will always give 5, as the last digit., , Hence the unit digit will be 5., , 36. (c) The last digit of the, expression will be same as the, last digit of 293+ g222,, , Now the last digit of 2°8is 6 and, the last digit of the 8222 is 4., , “ 6+4=10., , Hence, unit digit = 0, , 37. (c) Find the last digit of 932”, , But 9927 = 932« 32432... < 32 times, , ga” = Q4x8x2~02, 31 times), , > Q2" = Qa, , where n= 8 x (32 x 32. - x 32 times), , , , Again 2*= (16)" => unit digit is 6, for, , everyne N, Hence, the required unit digit = 6, 38. (a) Sum of square natural, number = 2(+1)(2n+1), Here, n = 100, ~ 100x101%201, = SE = 338350, , Then, Unit digit =o, , R;, kesh Yaday Readers Publication Pvt. Ltd., , Ils, , Sa