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we, , <5 > Sap ee., , Ex.44: Which of the following, , completely divide (341 + 722, (a) 4 (b) S2. (c) 17 (da) ®), , , , » the divisor is, ient and -5, , nder. If the, , , the dividend is:, (a) 808 (b) 5008, (c) 5808 (d) 8508, , The divisor is 321, the quotient, 11 and the remainder 260., Find the dividend., , (a) 3719 (b) 3971, , (c) 3791 (d) 3179, , In a division sum, the divisor, is 5 times the remainder and, the quotient is 6 times the, remainder which is 73. What, , , , , r vy (a +p) ULVv15}, , , , , , , , , , , , , , , , , , by (ab) (a +b)fa - b) mined, (a8) 8) | eB) aed v, (atb) (a-ab+B) | @=b) a**b*+ab)| [arb fab) | ath, , , , , , EXERCISE, , is the dividend ?, , (a).169943 (b) 159963, (c) 159943 (d) 159953, The, , sum of 20 odd natural, number is equal to :, , (a) 210 (b) 300 (c) 400 (d) 240, When a number is divided by., 56, the remainder obtained is, 29. What will be the remainder, when the number is divided by, 8?, , (a)4 ) 5 (3 7, , A number when divided, successively by 4 and 5 leave, the remainder 1 and 4, , Rakesh Yadav Readers Publication Pvt. Ltd., , respectively. When it ; 18., successively divided by 5 an *, 4 the respective remainder, will be:, , (a) 4,1 (b) 3,2 (c) 2,3 (d)1,2, , 7. 4°1+462+453+464 is divisible by:, (a) 3 (b) 10 (c) 11 (@)13, , 8. (375 + 376 + 327 + 378) is divisib, by: YY, (a) 11 (b) 16 (c) 25 (d)30, , 9. The least number, which mv, , 1, be added to 6709 to make ‘, exactly divisible by 9, is c, (a)5 (bh) 4 (ce) 7 (d)2 (, , 2%

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ri, , b, ae, , y 10., , If 78*3945 is divisible by 11, where * is a digit, then *, equal to :, al 0 (3 (d)s, . When a number is divided by, 357 the remainder is 39. If, same number is divided by 17,, the remainder will be :, (a—)9 (b)3 (ce) 5, , is, , (a) 11, , . 12, A number when divided by 6, , leaves remainder 3. When the, square of the same number is, divided by 6, the remainder is:, 0 1 (2 «3, , 13. When a number is divided by, , R, th, , 1, , 893, the remainder is 193,, What will be remainder when, it is divided by 47 ?, fa)3 (0) 5 (ce) 25 (ay 33, 4. Anumber divided by 13 leaves, a remainder 1 and if the, quotient, thus obtained, is, divided by 5, we get a, remainder of 3. What will be, the remainder if the number, is divided by 65 ?, , (a) 28 (b) 16 (c) 18 (a) 40, , 15. Which of the following number, , is NOT divisible by 18 ?, (a)54036 (b) 50436, (c) 34056 (d) 65043, , 16. If n is an integer, then (n3 - n), , 17. A 4 digit number is formed b, , is always divisible by:, 4 5 (6 (7, Tepeating a 2 digit numbe, such as 2525, 3232, etc. Any, number of this form is always, exactly divisible by: , _ 4, (a)7 only (b) 11’On i,, (c) 13 only (@) Smallest 3, digit prime number wy %, , , , , , st ? divided by the ivisor, the, ndt Temaind: ‘spectively 3, and 4. If the“éum of the two, )1, RUmbers beydivided by the, py 84Me divisor, the remainder is, 1) 2: The divisor is :, ‘sh, 12 7 (5 (a3, a 190A number when divided by 5, 30 leaves remainder 3. What is, the remainder when the, at “Ware of the same number is, divided by 5?, \2 f1 a2 (c) 3 (a4, , , , , , 20., , 21,, , 22., , 23., , 24,, , 25,, , 26., , 28., , 29., , 30., , 31., , If the number 4 8 3 27* Bis, divisible by 11, then the, , missing digit (*) is, , (aS (b) 3 (c) 2 (djl, , A number, when divided by, 136, leaves remainder 36. If, , the same number is divided by 33., , 17, the remainder will be, fa)9 (7 (3 Wa, Two numbers, when divided by, 17, leaves remainder 13 and 11, respectively. If the sum of, those two numbers is divided, by 17, the Temainder will be :, (a) 13 (b) 11 (cy 7 (a4, , A number, when divided by, 221, leaves a remainder 64., What is the remainder if the, same number is divided by 13?, (@)0 (bh) 1 (ey 11 (a) 12, When a number is divideds b, 387, the remainder obtai, 48. If the same numbe, divided by 43, the remaindér, obtained will be 2.4, “, , , , , 0 ()3 (5 (d) 35, , When two numb rare, Separately divided By 33, the, remainders, re j21 and 28, , , , respectively.), , , , a) 9, , (b) 10, 2%.When 'n' is divisible by 5 the, , (c) 11 (13, Femainder is 2. What is the, remainder when n? is divided, by 5?, , (a)2 (b) 3) () 1 a4, , A number when divided by 49, leaves 32 as remainder, The, number when divided by 7 will, have the remainder as:, , (a)4 () 3 (2 (as, When a number is divided by, 36, the remainder is 19. What, will be the remainder when, the number is divided by 12 ?, (a)7 (b)5 (c)3 (do, , In a division sum, the divisor, is 10 times the quotient and 5, times the remainder. If the, remainder is 46, then the, dividend is :, , (a) 4236 (b) 4306, , (c) 4336 (d) 5336, , When a number is divided by, 24, the remainder is 16. The, remainder when the same, , Fakesh Yadav Readers Publication Pvt. Ltd., , 32., , 34,, , 35., , , , , , , , 36:, , 37., , 38., , 39., , 40., , 41., , 7, , 43., , Negor, , , , number is divided by 12 is, (a)3 (b)4 (c) 6 (d)8, The expression 8" - 4", where, n is a natural number ‘is, always divisible by, , (a) 15 (b) 18 (c) 36 (4) 48, (4°! +4° +463) is divisible by, (a) 3 (b) 11 (c) 13d) 17, , When an integer K is divided, by 3, the remainder is 1, and, when K + J is, divided by 5, the, , , , 63” (c) 64 (d)65, A number, when divided by 91, gives a remainder 17. When, th€’same number is divided by, (13, the remainder will be :, yO )4 (6 @3, A’number when divided by 280, leaves 115 as remainder. When, the same number is divided by, 35, the remainder is:, , (a) 15 (b) 10 (c) 20 (d) 17, , A certain number when divided, by 175 leaves a remainder 132., When the same number is, divided by 25, the remainder is:, (6 )7 (8 (a9, Which one of the following will, completely divide by 57! + 57 + 573, (a) 150 (b) 160 (c) 155 (d) 30, Which of the following numbers, will always divide a six-digit, number of the form XY xy xy, , , , (where 1 <x<, l<y <9)?, (a) 1010 (b) 10101, (c) 11011 (d) 11010, , A positive integer when divided, by 425 gives remainder 45., When the same number is, divided by 17, the remainder, will be, , (a5 (2 (11 @13, , A number x when divided by 289, leaves 18 as the remainder., The same number when divided, by 17 leaves y as a remainder., The value of y is, , (a)5 (b)2) (3 (d)1, , 42. When n is divided by 6, the, , remainder is 4. When 2n is, divided by 6, the remainder is:, (a)2 (0 (4 @i, , In a division sum, the divisor, is 3 times the quotient and 6, times the remainder. If the, , remainder is 2, then the, , dividend is ; ., , (a) 50 (b) 48 (c) 36 (a) 28, webs: alll 25,

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44, In a divison sum, the divisor, is 12 times the quotient and 5, times the remainder. If the, remainder is 36, then the, , dividend is :, (a) 2706 (b) 2726, (c) 2736 (d) 2262, , . For any integral value of n, 37°, + 9n + 5 when divide by 3 will, _ leave the remainder, , (a) 1 (b) 2, 1 (e) 6. (ec) 11. (c), 2. (ce) 7. (bd) 12. (d), 3. (ce) 8. (d) 13. (b), 4. (c) 9. (a) 14. (d), 5. (b) 10. (d) 15. (d), , , , 1. (c) Remainder = 48, , Divisor = 48 5 = 240, 240, i = ——=24, Quotient To, Dividend = 240 x 24+ 48, = 5760 + 48, = 5808, , 2. (c) Dividend = Divisor 4, Quotient + Remainder, = 321 x 11 +260, = 3531 + 260=3791 *, 3. (c) Remainder = 7& @ >, Quotient = 6x73 Aw, Divisor = 5x73, , , , Dividend = +73, 9943, 4. (c) 1, 3, 20th term, a=I1,d= 20, , n, sum = 2 [2a + (n - 1)d], , _20, =F [2 x1+(20-1)2], , = 10[2x1 + 19 x2] = 400, Alternate :, , The sum of first n odd natural, numbers = n? = 20? = 400, , , , , , , , (c) 0 (d) 5, 46. The quotient when 10'” is, divided by 5’ is : 49., (a) 10% (b) 27°., (c) 275* 1025 (d) 275« 1075 50., 47, The remainder obtained when, 23% + 319 is divided by 54 51., (a) 0 (b) 1, _ (e) 3 (d) C.N.D, 48. (195 + 215) is divisible by, (a) Only 10, (b) Only 20, ANSWER KEY, 16. (c) 21. (d) 26. (b) + (b), 17. (d) 22. (c) . 27. (d) - (d), 18. (c) 23. (d) 28. (a) . (a), 19. (d) 24. (c) 29. (a) » (c), 20. (d) 25. (d) 30. (d) . (b), ed, SOLUTION, 5.) Rema ee, 8, Re 5, 6. {ey i divided, “Remainder 9., 437 1, 5/9 4, , 9x4+1= 37, Number is 37, S137 2, Ale. 3, , T, Remainder is 2, 3, (b) 451+ 452 + 459+ gor, = 491/49 + 41 +4 42 + 43), = 491(1+4+ 16+ 64), , 10., , , , n, , = 45x 85, = 4x 4x 85, , = 4x 340 11., = 4x 34x 10, , Now, check with option 12, , Only, check with option, , Only 10 can divide this., , (d) (35+ 376+ 377+ 378), 395 (3°+ 31+ 32+ 39), , Rakesh Yadav Readers Publication Pvt. Ltd., , (c) Both 10 & 20, , (d) Neither 10 nor 20, , If (17)*! + (29)*' is divided by 23, Find the remainder, (a)1 (6 (c)O i2, If (3)*" + (7) always divisible, (a) 10 (b) 49 (c) 52 (dag, If m® - n™ = (m + n); (m, y, e prime numbers, then wha, can be said about m and n:, , (a)m, nar nly even integer, (b) m, p ‘are*enly odd integers, , (b) mis’ é nd n is odd, , (dn ese, , ~, , 36. (b) 41. (d) 46. (c), , 37. (b) 42. (a) 47. (a), , 38. (c) 43. (a) 48. (c), , 39. (b) 44. (c) 49. (c), , 40. (c) 45. (b) 50. (c}, 51. (ce), , = 3%(1+3+9+27), , = 3% x 40 = 3% x 120, Now, check with option, Only, check with option, Only 30 can divide this., (a) 6709, , => 6+7+0+9=22, , [9 - (divisibility property), Sum of digits must be divisilf, by 9], , So 22 + 5 = 27 is divisible by’, 5 is answer, , —, @ 78 * 3945, os eae ry, , Odd place: 7+*+9+5=21+, Even place :8+3+4=15, (21+ *) -(15)= either 11 or 0, (21 +*)-15 = 11, , 21+*=26, i's, Remainder of number _ 39, (c) Remainder of number = 5, 17, , = remainder = 5, (d) Shorcut Method, , Let number is: 9 (Gives rem), der 3 when divided by 6), , 2, 2 2 & _ Remainder’, , 6 6, i, I?, , 2, , Now

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Remainder ofno. _ 193 22. (c) (dividend = divisor x quotient 2 7 4g, 3. () ~ 47 47 + remainder) r = 3 7 5 = remainder= 4, = remainder = 5 First no. = (17 x n) +13, t 'n' = remainder ofno. 32, @) 13 1051 Let 'n'= 1 28, (a) “maim ->, “s/s 3 = (17 1) +13 7, TT = 30 => Remainder = 4, 5x1+3=8 Second no, = (17 «n)+11 fa) remainder of no. 19), 13x8+1= 105 =(I7x1)+11= 28 12 12, remainder = 105 + 65 According to question => remainder 4 7, Remainder = 40 ap 30. (d) A, +28 58 % A 4, . (d) A number will be divisible by eS a, 2S . - Quotient a7 : Remainder, 18 if it is divisible by 2 and 9 17 17 > temainder = 7 1 :, Clearly we can see 65043 is not Alternate:- 1, divisible by 2. Because unit Divisor =, digit of 65043 is 3 so this will Réttidindes eae es 2, not be divsible by 18 Wa ide er [2s [23, . (o (n° - n) and n is any integer. ° 11 ~ Remainder 3 230 46, , put n= 2 so, 27>-2=6, , It will be always divisble by 6, (Put n = 2,3,4...), , (d) Smallest 3 digit prime number is '101', , ayzy is always divisible by 101, Hence, 101 Will be the divisor., {c) Shortcut Method, , divisor = Remainder 1 +, Remainder 2 - Remainder 3, =3+4-2=7-2=5, , 23., , 25., , (d) Let no. be 8, #2 1)+28=61, aa ect, > 5 ccording to question 33., = 4 remainder, tmate:- +61 115, 33 > 33, (Remainder)’ = 16 Remainder aa, Alternate:- 4, Divisor = Remainder 1 +, , , , +*=10+* 26., ould be either, 710 0 11429733 cece etc., 222-110 +") = 11, 22-104. 11, 12-*= 1, ey 27., , 4 “etander of no,, , . ndaag =2, , 's], NYaday Readers Publication Pvt. Ltd., , ~ (c), , Remainder 3 = 24 - 17 = 7, , Re, (a) Semainder of no. nance of no., , = remainder =, , Remainder of oy, 43 43, , > =, (d) first nOh\={(3%y n) + 21, Let nog=, tD 54, ne = (33 x n) + 28, , , , Remainder 2 - Remainder 3, 33 = 21 + 28- Remainder 3, Remainder 3 = 16, (b) (27 + 27 + 273 + 274), = 271(29 + 2! + 2? +23), =27(1+2+4+8), = 27x 15 = 27x30, It is divisible by 10, , 35., (d) & = remainder 2, 5, , If we put n = 7 Then it satisfies, above situation, Son=7, , 36., , expression = 8?, o = ( Divisor x Quotient), + Remainder, , = (230 x 23) + 46= 5336, , 16, “72, , p) Remainder Remainder of no. _, - ae, = 4 is a Tae, - (d) 87n= 1 Biflssencne:, tal number), Put, n= 2,, , (n is a natu, = 64-16=48, . 8° - 4" is divisible by 48, 48 is completely divisible by 4, so f is divisible 4, (a) (451 + 492 + 459), = 45/49 + 414 42), =45(1+4+16) = 45x27, Now check the options, Only 3 divides it. So '3' is answer, (c) Always do these types of, question by options to save time, Pick up the option and follow, the question instruction, take option (c), 64 = Divide 3 it gives remainder 1, Now add 1 to 64, , 65, “5 > Temainder '0' it satisfies, , So, k = 64 this is answer, , (b) Remainder of no. 17, SSS es os, 13 13, remainder = 4, (b) Remainder of no. _ 1s, , 35 35, , Remainder = 10, , Jl27

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PT, , Remainder ofno. _ 132, 37. (>) 25 ~ 25, remainder = 7, (ce) 5+ 572 + 573, = 571 (59+ 5! + 54), = 57 (1+5+ 25), = 571 x 31 = 57x 155, Check with option,, So 155 is answer, 39. (b) Number = xyxyxy, = xy x 10000 + xy x 100 + xy, = xy (10000+ 100 +1), = xy(10101), Hence, option (B) will divide answer, Alternate:, You can assume (121212,, 343434. .) any number divisible by option, So that number, is divisible by exactly that's the, answer, Remainder of no. _ 45, 40. (¢} 17 SST, = remainder = 11, Remainder ofno. _ 18, 41. (d) nn | -i7, = remainder = 1, 42. (a) @ = remainder 4, 10, Ifn=10> 6, , Note : Always put aA. 1, , = remainder = 4 (matched) n =, 20, an=2*105—, , = remainder = 2, , of questions, a, , 43., , 44., , 45., , Rakesh Yadav Readers Publication Pvt. Ltd., , (a), Remainder : Divisor : Quotient, 3): 1, 1 6, 1 6: 2, bob F, Actual —>2 12 4, Dividend = (Divisor < Quotient), + remainder, = (12 x 4)+2=50, (c), Remainder :, 1 5, 12, x3 IN, Dividend = (divisor < a, mainder, = (180 x 15) +, = 2736, (b) a, , Qh = 2, , = remainder = 2, , , , 57, , Note: value of n can be 1,2,3,4,, , J (c) 10 + 575, , 100,100, , =7100, 525925 975 525, , = 275% 1025, , 47., , 48., , 49. (HA ( ,, Bute s teat a, visor: Quotien sible (a + b), , 51., , (a) We know that (a" + b*) ig, always divisible (a + b) then,, where n - odd power, , 233 + 31°) is Always divisible, , by (23 + 31) =, So remainder is ee, (c) (a® + b*), is always, , divisible by (a + b), When n -> odd power, , (19+ ca, , Factor of, 1, 2, 4, 5, 10, 20,, 40) f Mei by (195 + 214, the Ins 10 & 20 is, divisible, , an + b), is always, , en n is odd power, ‘hen,, , (174 + 29%) is always divisible, by (17 + 29) = 46, factor of 46 (1, 2, 23, 46), So, (174! + 29%) is perfectly, divisible by 23, hence, Remainder '‘O', , (c) 3° +72, Equalising the power, 341+ (72) = 34+ 494, , 341 + 49%! is always divisible, (3 + 49) = 52, , So 52 is divisible by (3*! + a, (c) m=-n™=min, Consider m = 2 and n =, then, , 25-S?=5+2, , 7=7, , Thus option (a) and (b) af, wrong and option (c) i, correct., , ———S ee., , Mi