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Number of zeroes or Number of, trailing zeroes, Number of zeroes, Eg. 1. 1500, , 2, 0, , 2. 1005, , , Expression, of zeroes:–, , 2×2×5×5, , Trailing zeroes, , zero, , , , , , Trailing zeroes/ number, , , , , , , , Expression, number of zeroes, expression, , Trailing zeroes/, , Examples–: 1., Expression, (i) 2 × 3 × 4 × 5, (ii) 22 × 24 × 25 × 30, (iii) 125 × 128 × 55, (iv) 3075 × 1024 × 512 × 216, (vi) 600 × 500 × 200 × 25, (vii) 100 × 300 × 400, (viii) (125)25 ×(128)22, (ix) (3075)5 × (1024)7, (x) 1000 × 5000 × 2500 × 6400, (xi) 103 × 254 × 643 × 125 × 32, soln:– (i) 2 × 3 × 4 × 5, , Expression, (pair), = 1 Ans., , , , , , , , , , , , Expression, zeroes:–, , Trailing zeroes/ number of, , Sol :- (ii) 22 × 24 × 25 × 30, n, , = 2 ×11 × 2 × 2 × 2 × 3 × 5 × 5 × 5 × 2 ×3, = 25 × 32 ×53 ×11, ,, =3, , , , =3, , Sol :- (iii) 125 × 128 × 55, = 53 × 27 × 5 ×11, n, , ,, =4, , , , =4, , Soln:- (iv) 3075 × 1024 × 512 × 216, = 5 × 5 × 3 × 41 × 210 × 29 × 23 × 33, = 222 × 34 × 52 × 41, ,, =2, , , Wizard of Maths "Rakesh Yadav Sir", , =2, , RAKESH YADAV READERS PUBLICATION PVT. LTD, , 1
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= 103 × 58 × 218 × 53 × 25, = 223 × 511 × 10 3, = 11, = 11+, , , 2., 125 = 5 × 5 × 5 = 53, , 3, , = 14, , 213, , soln:– (1 × 3 × 5 × 7 ×....... 99) × 213, = 10 + 2 = 12, , =8, , , , =8, , , Sol :- (vi) 600 × 500 × 200 × 25, = 2 × 3 × 102 × 5 × 102 × 2 × 102× 52, n, , 5, 15, , = 22 × 3 × 53 × 10 6, , 25, , =2, =2+, , , , 6, , 35, , =8, , 45, , Soln:- (vii) 100 × 300 × 400, = 102 × 3 × 102 × 4 × 102, = 3 × 4 ×106, , , Sol :- (viii), n, , 5 5, , 55, , [am × an = am+n], , 65, , =6, , 75, , (125) × (128)22, = (53)25 × (27)22, = 575 × 2154, 25, , 5 5, , 85, 95, , ,, , 10 2, = 75, = 75, , , , 3. 3125 × 16, Sol :- (ix) (3075) × (1024), = (3 × 5 × 5 × 41)5 × (210)7, = 35 × 510 × 415 × 270, n, , 5, , 7, , ,, , = 55 × 2 4, = 5(54 × 24), = 5×(5×2)4, = 5×(10)4, , = 10, , , , Soln:- (x) 1000 × 5000 × 2500 × 6400, = 103×5×103×52×102×26×102, = 2 6 × 53 ×, , =4, , , , = 10, , 4. 37 × 65 × 73 × 97 × 55 × 40, , 5×11 23 ×5, , 5×13, , 10 10, , =3, , , , ,, , 5., , 1 × 3 × 5 × 7 × 9 ×..... 999, , =3, , , , = 3+, , 10, , = 13, , Soln:–, , =0, , Expression, , (xi) 103 × 254 × 643 × 125 × 32, = 103 × (52)4 × (26)3 × 53 ×25, , 2, , RAKESH YADAV READERS PUBLICATION PVT. LTD, , Wizard of Maths "Rakesh Yadav Sir"
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6! = 6 × 5 × 4 × 3 × 2 × 1 = 720, , 6. 3200 × 1000 × 40000 × 32000 × 15000, , 50!, 50! = 50 × 49 × 48 × ....... ×1, = 50 × 49!, = 50 × 49 × 48 × 47!, , = 32 × 102 × 103 × 4 × 104 × 32 × 103× 15×103, = 25 × 22 × 25 × 3 × 5 × 10 15, = 1+ 15 = 16, , , , Approach:–, , A special Implication:–, 4, zero, e.g, , Factorial, , 1., , no. of 5's, , Factorial, , product, , no. of zeroes, = 1,, , no. of zeroes, , end, , zeroes, , count, , = 1×2, 3! = 1 × 2 × 3 × 4, , factorial 5!, 6!, 7!, 8!, 9!, no. of zeroes, , maximum, combination, , , 11!, 12!, 13!, 14!,, , Ex–, factorial, , 5! = 5 × 4 × 3 × 2 × 1, no. of 2's = 3, no. of 5's = 1, no. of pair (2 & 5) = 1, , form, , , , no. of zeroes = 1, , 3., , no. of zeroes, , factorial, 2', , s, , no. of zeroes, , (Repetition), , equal, , Integer, , no. of zeroes, , Integer, , no. of zeroes = 25 + 5 + 1 31, , 5's, , factorial, , :-, , no. of zeroes 3, , 1. Integer, , N, , N, , prime factor, , no. of zeroes, , no. of zeroes = 24 + 4 = 28, , Soln:-, , prime factorization, , 's, , no. of zeroes, , Ex:-e N!, , no. of, , zeroes, , no. of zeroes, , and 125!, , product, , 5x, n!, , Factorial, , n, n, n, n, 2 3 .... x n 5 x , 5 5, 5, 5, Count trailing zeroes / no. of zeroes in factorial, , Ex– 350!, , of numbers., Factorial:–, , , , 350 350 350, 2 3, 5, 5, 5, , Factorial, , = 70 + 14 + 2 = 86, Factorial, , n, e.g., , n! or, , n, , , , or n! means = n × (n–1) × (n–2) × (n–3) ×..... ×1, , = 86, , 2., , 5! = 5 × 4 × 3 × 2 × 1 = 120, , Wizard of Maths "Rakesh Yadav Sir", , RAKESH YADAV READERS PUBLICATION PVT. LTD, , 3
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Ex– 350!, , 5 350, 5 70, 5 14, 2, , , , no. of zeroes., , , 6., , Integer, , factorial, , 1×2×3×4×..... 100, , Soln:- 1×2×3×4×..... 100., = 100!, , prime factor, , 2, 2, 2, 2, 2, 2, , Ex. 500!, , 7 500, 7 71, 7 10 no. of 7's = 71+10+1= 82, 1, Examples, 50!, , Soln:-, , 5 50, 5 10, 2, , 3., , = 12, , Soln:-, , = 50+25+12+6+3+1 = 97, 5, , = 24, = 24, , 2, , 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, , = 26, , 600, 150, 30, 6, 1, , = 11, , 5×5, , 5×5, , 5, , = 12, = 11, = 11, , , = 187, , , 4., , 2, , 1 × 3 × 5 × 7 × ....× 99 × 2048, Soln:- (1 × 3 × 5 × 7 × ....× 99) × 2048, 2048 = 211, , 600!, , 5, 5, 5, 5, , = 20+4 = 24, , 7., , 5 110, 5 22, 4, , , , 5 100, 5 20, 4, , , , 110 !, , Soln:-, , 100, 50, 25, 12, 6, 3, 1, , 5, 2, , , 2., , no. of zeroes = 112, , no. of zeroes = 70 + 14 + 2 = 86, , NOTE:, , 1., , = 112, , 2145 × 5234, , Soln:- 2145 × 5234 = 2145 × 5145 × 589, = (2×5)145 ×589, = (10)145 × 589, , , 5., , = 145, , 212 × 512, , 8., , 2 × 4 × 6 × 8 ..... × 360, , Sol :- 2 × 4 × 6 × 8 ..... × 360, = 2 × 1 × 2 × 2 × 2 × 3 × 2 × 4 × ..... 2 ×, 180, = 2180 (1 × 2 × 3 × 4 × ..... × 180), = 2180 × 180!, , Sol :- 212 × 5212, = 2112 × 5112 × 5100, = (2×5)112 × 5100, = (10)112, , n, , n, , 4, , RAKESH YADAV READERS PUBLICATION PVT. LTD, , 5, , = 36 + 7 + 1 = 44, , 5 180, 5 36, 5 7, 1, , Wizard of Maths "Rakesh Yadav Sir"
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= 44, , , 9., , 4 × 8 × 12 × ......×480, , = 25+5+1+3, , Soln:- 4 × 8 × 12 × ...... ×480, = 4 × 1 × 4 × 2 × ..... 4 × 120, = 4120 (120!), = (22)120 × (120!), = 2240 × 120!, , , , 5, , 14., , = 34, 21!, , , Soln:, , 5 120, 5 24, 4, , = 28 Ans., , (5 × 10 × 15 ×..... 500), , = 24, , =4, , = 24 - 4 = 20, 55 × 56 × 57× ...... ×101, , Soln:-, , 55 × 56 × 57× ...... ×101, = (1×2×3×....101) - (1×2×3×....54), , 101! ÷ 54!, =, , = 50 + 25 + 12 + 6 + 3 + 1 = 97, , = 24, , , , 11., , 2 100, 2 50, 2 25, 2 12, 26, 2 3, 1, , = 97, , , (8125 - 8124, , Soln:-, , = (8125 - 8124 - 8123) (3225 - 3224 - 3223), = 8123 (82 - 8 - 1) 3223 (32 - 3 - 1), = 8123 × 55 × 3223 × 5, = 8123 × 8123 × 11 × 52, >, =2, , = (10 × 1 ×10 × 2 ×10 × 3 × 10 × 4× .... 10 × 70), = 1070 (1 × 2 × 3 × 4 × ..... 70), , , , (14 × 24 × 34 × 44 × ..... × 1004), , Soln:- (14 × 24 × 34 × 44 × ..... × 1004), = (1 × 2 × 3 × 4 × .....×100)4, , = 70 +14+2, = (100!), , = 86, 12., , =2, , , 17., , 5 70, 5 14, 2, , = 12, , = 24 - 12, = 12, - 8123 ) (3225 - 3224 - 3223), , 16., , (10 × 20 × 30 × 40 × .... 700), , = 10 70 × 70!, , [am ÷ an = am-n], , 15., , = (5 × 1 × 5 × 2 × 5 × 3 × ..... 5 × 100), = 5100 (1 × 2 × 3 ×... 100), = 5100 × 100!, 2, , 100!, , 100! ÷ 21!, , = 24 + 4 = 28, , , 10., , 3, , 125!×5, , =, , 5 100, 5 20, 4, , 4, , 78! + 79!, = 78! + 79 × 78!, = 78! (1+79), , 5 78, 5 15, 3, 13., Soln:-, , 126! - 125!, 126! - 125!, , = 126 × 125! - 125!, , 18., , Expression, a × b × c.....× z, , 5 125, 5 25, 5 5, 1, , = 125! (126-1), = 125! × 125, , Wizard of Maths "Rakesh Yadav Sir", , a = 13, b = 24, c = 35......., z = 2628,, Soln:– a × b × c.....× z, = 13 × 24 × 35 × 46 × 57 × 68 × ......× 2628, , (5×5)27 = 554, , RAKESH YADAV READERS PUBLICATION PVT. LTD, , 5
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= 7 + 12 + 17 + 22 + 54 = 112, = 112, , , 19., , Soln:–, , n, , 10n, , 150 !, 10n, 150!, , = 30+6+1 = 37, zeroes, , n, , n, , 20., , Soln:-, , n, 200!, , = 37., 3, , n, , n, , 200 !, 3n, , 3, 3, 3, 3, , 200, 66, 22, 7, 2, , 200!, , = 66 + 22 + 7 + 2 = 97, n, , 6, , RAKESH YADAV READERS PUBLICATION PVT. LTD, , Wizard of Maths "Rakesh Yadav Sir"
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Exercise, 1., , 40!, (a) 9, (c) 10, , 2., , (5 × 15 × 25 × 35 × ...×95)×29, (b) 10, (d) 15, , 33 × 44 × 55 × ..... ×4949, (b) 215, (d) 300, , 6!×13!×518, (a) 14, (c) 12, , (b) 18, (d) 10, , (a) 38, (c) 49, , (b) 39, (d) 35, , (a) 14, (c) 57, , (b) 9, (d) 10, , (a) 16, (c) 40, , (b) 38, (c) 39, , 19., (b) 22, (c) 20, 20., (b) 30, (c) 28, 21., , (b) 190, (d) 194, , 22., , 10! + 100! + 1000! + 1000000!, , (5 )!, 10, , 510 1, 4, , 59 1, 4, , (b), , (d), , 1 510, 4, , 157!, , 510 1, 4, , (a) 38, (c) 37, , (b) 3, (c) 6, , (a) 124, , (b) 97, , (c), , (d) 24, , 120, , N!, , (b) 4! + 9!, (d) 8!+12!, , 10n, , (a) 2, (c) 4, 23., , 11! × 22! × 33! ×....×1010!, (a) 5! + 10!, (c) 6!+11!, , 13., , 18., , (226 × 227 × 228 × ....×1000), , (c), , 12., , (b) 10, (d) 15, , (a) 200, (c) 250, , (b) 174, (d) 178, , (150!–31!), , (a), , 11., , (1 × 3 × 5 × 7 × .....×99)×211, , (a) 9, (c) 19, , (b) 222, (d) 444, , (238! – 120!), , (a) 196, (c) 192, 10., , 16., , (b) 18, (d) 12, , (a) 32, (c) 34, 9., , (b) 111, (d) 113, , (62! + 78! + 94!), , (a) 28, (d) 24, 8., , (b) 13, (d) 10, , (a) 12, (c) 11, , 17., , (a) 16, (c) 14, 7., , 15., , 50! + 60! + 70!, (a) 12, (c) 15, , 1501!, (a) 372, (c) 375, , 6., , (b) 24, (d) 22, , (b) 78, (d) 90, , 899!, (a) 111, (c) 333, , 5., , 14., , 450!, (a) 112, (c) 110, , 4., , (b) 8, (d) 12, , 100!, (a) 25, (c) 23, , 3., , (a) 60, (c) 80, , (a) 59 and 55, , n, (b) 39, (d) 40, , (c), 25., , Wizard of Maths "Rakesh Yadav Sir", , 26., , (b) 54 and 50, , 74 and 70, , (d) N.O.T, , 100 × 99 × 98 × 974 ×.... ×1100, 1, , (a) 1000, (c) 1300, , (1! × 2! × 3! × 4! × 5! ..... ×29!), ?, , N, , n, , 2, , 3, , (b) 1200, (d) 1124, n, , RAKESH YADAV READERS PUBLICATION PVT. LTD, , 7
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40., (a) 6, (c) 8, 27., , 28., , (a) 15, (c) 16, 34., , (a) 134, (c) 138, 39., , 3200 ×1000 × 1500 × 64000 × 1200 × 16000, (b) 1017, (d) 1022, , 10! 20!, 10! 20!, (b) 2, (d) 5, , 124! 125!, 125!124!, (a) 1058, (b) 1028, (c) 1030, (d) N.O.T, 3200 + 6400 + 4500 + 6800 + 3600 + 5500+70000, (b) 104, (d) 106, , 100! 200!, 100! 200!, (a) 1073, (c) 1097, , 48., , (b) 1024, (d) 1049, , 10 × 30 × 50 × 70 × 90 × .....× 990, (a) 99, (c) 50, , (b) 40, (d) 25, 49., , (b) 135, (d) 124, , 1050 1000, , (b) 100, (d) 25, 20, , 105 1000, (a) 110, (c) 60, , 2, , (b) 160, (d) 120, , 33×44×55×....×4949, (a) 200, (c) 300, , 8, , 47., , (b) 25, (d) 25, , (100 × 200 × 300 ×....×6000), , (b) 2, (d) 1, , (a) 102, (c) 105, , 15 × 25 × 35 ×.....×455, (a) 10, (c) 50, , 38., , 46., , (b) 19, (d) 0, , 5 × 10 × 15 ×......×105, (a) 21, (c) 18, , 37., , (b) 26, (d) 25, , 9 × 18 × 27× ...... ×729, (a) 20, (c) 0, , 36., , (b) 224, (d) 124, (b) 12, (d) 20, , 15000+10000+1500+150, , (a) 6, (c) 3, , 6 × 12 × 18× ..... ×630, (a) 24, (c) 27, , 35., , 44., , 45., , 3 × 6 × 9×.... ×210, , (b) 1, (d) 3, , (a) 1016, (c) 1018, , (b) 9!, (d) 10!, , 1 × 2 × 3×.....× 900, (a) 225, (c) 200, , 33., , (b) 300, (d) 375, , 39! × 511! × 27!, (a) 11!, (c) 7!, , 32., , 43., , (b) 38, (d) 40, , (2123 – 2122 – 2121) (3234 – 3233 –3232), , (a) 10, (c) 4, , 144! × 5 × 15× 22 ×11 × 44×135, (a) 37, (c) 39, , 31.., , 42., , (b) 8, (d) 10, , 1142! × 348! × 17!, , (b) 175, (d) 76, , (a) 0, (c) 2, n, , (a) 372, (c) 371, 30., , 41., , (b) 2, (d) 0, , n, (a) 7, (c) 9, , 29., , (a) 173, (c) 249, , (b) 12, (d) 9, , (8253 – 8252 – 8251) (3221 – 3220 – 3219), (a) 1, (c) 3, , 311×312×315×....×1000, , (b) 250, (d) 150, , RAKESH YADAV READERS PUBLICATION PVT. LTD, , Wizard of Maths "Rakesh Yadav Sir"
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50., , 25! 32! 45!, 25!32! 45!, (a) 1023, (c) 1017, , 51., , 53., , 54., , 55., , 5!5! + 10!10! + 50!50! + 100!100!, (b) 120, , (c), , (d), , 130, , 3 × 6 × 9 × 12 × ..... × 990, (a) 81, (c) 78, , 56., , 64., , (b) 13, (d) 12, , (b) 230, (d) 229, , 65., , 66., , 59., , 60., , (b) 104, (d) 109, , (b) 4, (d) 5, , Wizard of Maths "Rakesh Yadav Sir", , (b) 1324, (d) 1322, , (3123 – 3122 – 3121) (2121 – 2120 – 2119), (a) 0, (c) 3, , (b) 1, (d) 2, , (a) 0, (c) 24, , (b) 1, (d) 3, Expression, , 1 × 2 × 3 × 4× .....×79, (a) 6, (b) 8, (c) 9, (d) 7, 69., , 127 × 128 × 129 ×....×359, (a) 55, (c) 55, , 70., , 71., , (b) 54, (d) 56, , 512 × 513 ×.... × 1221, (a) 176, (c) 179, , 37 × 65 × 73 × 97 × 55 × 40, (a) 3, (c) 2, , (b) 770, (d) 778, , 12 × 23 × 34 × 45 × ..... × 101102, , 68., , 12 × 23 × 34 ×45 × 56 × ..... × 2930, (a) 105, (c) 106, , 12 × 23 × 34 × 45 × ..... × 7778, , 67., , zeroes., (b) 809, (d) 810, , (b) 362, (d) 363, , (a) 1325, (c) 1323, , 310 × 610 × 910 × 1210×..... × 99010, (a) 812, (c) 811, , 12 × 23 × 34 × 45 × 56 × ..... × 5354, , (a) 768, (c) 777, , 421 × 821 × 1221× ..... × 14821, (a) 231, (c) 228, , 58., , (b) 80, (d) 82, , (b) 48, (d) 49, , (a) 360, (c) 361, , 4 × 8 × 12 × ..... × 148, (a) 10, (c) 11, , 57., , 63., , (b) 10469, (d) 10471, , (a) 100, , 1 × 2 × 3 ×..... ×198, ,x, 5x, (a) 47, (c) 50, , (b) 14, (d) 16, , (20! × 21! × 22! .....× 33!)3!, (a) 10468, (c) 10470, , 62., , (b) 51, (d) 50, , Expression, , n, , (a) 12, (c) 15, , (a) 52, (c) 53, , (b) 12, (d) 1, , n!, , 1 × 2 × 3 × 4 × 6 × 7 ×..... ×108, ,x, 3x, Expression, , (b) 1020, (d) Not, , 211 × 1253 × 711 + 31 1× 711 × 210 × 51 +1112 × 213 × 5125, (a) 9, (c) 13, , 52., , 61., , (b) 178, (d) 177, , 1200 × 7000, (a) 4, (c) 5, , (b) 3, (d) 6, , RAKESH YADAV READERS PUBLICATION PVT. LTD, , 9
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72., , 1200 + 7000, (a) 4, (c) 3, , 73., , 1020 × 1040, (a) 55, (c) 58, , 74., , (b) 167, (d) 164, , 1500 × 1200 × 800 × 500, (a) 8, (c) 14, , 10, , (b) 14, (d) 15, , 5 × 10 × 15 × 20 × ...... × 850, (a) 165, (c) 166, , 80., , (b) 123, (d) 125, , 5 × 10 × 15 × 20× ...... × 85, (a) 13, (c) 16, , 79., , (b) 100, (d) 97, , 10 × 20 × 30 ×...... × 1000, , (a) 122, (c) 124, 78., , (b) 21, (d) 20, , 10 × 20 × 30 ×...... × 790, (a) 99, (c) 98, , 77., , (b) 18, (d) 21, , 10 × 20 × 30 × 40 × 50 ×...... × 150, (a) 19, (c) 18, , 76., , (b) 56, (d) 60, , 1020 + 1030, (a) 20, (c) 19, , 75., , (b) 2, (d) 1, , (b) 10, (d) 12, , RAKESH YADAV READERS PUBLICATION PVT. LTD, , Wizard of Maths "Rakesh Yadav Sir"
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ANSWER KEY, 1., 2., 3., 4., 5., 6., 7., 8., 9., 10., 11., 12., 13., 14., 15., 16., 17., 18., 19., 20., , (a), (b), (b), (b), (b), (c), (a), (a), (d), (d), (a), (a), (c), (a), (c), (a), (c), (a), (a), (b), , 21., 22., 23., 24., 25., 26., 27., 28., 29., 30., 31., 32., 33., 34., 35., 36., 37., 38., 39., 40., , (b), (a), (b), (a), (d), (b), (b), (b), (c), (a), (c), (b), (c), (d), (b), (c), (c), (a), (b), (a), , Wizard of Maths "Rakesh Yadav Sir", , 41., 42., 43., 44., 45., 46., 47., 48., 49., 50., 51., 52., 53., 54., 55., 56., 57., 58., 59., 60., , (b), (d), (b), (c), (c), (c), (d), (c), (c), (c), (d), (c), (a), (b), (a), (c), (a), (d), (c), (a), , 61., 62., 63., 64., 65., 66., 67., 68., 69., 70., 71., 72., 73., 74., 75., 76., 77., 78., 79., 80., , (c), (a), (b), (a), (b), (b), (b), (d), (d), (a), (c), (b), (d), (a), (c), (d), (c), (d), (d), (a), , RAKESH YADAV READERS PUBLICATION PVT. LTD, , 11
