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161. If the six-digit number 479xyz is exactly divisible by 7,11 and 13, then the value of {(y+z) ÷ x} is equal to
- A. 7/13
- B. 11/9
- C. 4
- D. 13/7
Answer: 4
Explanation: LCM of 7, 11 and 13 = 1001 Numebr 479xyz is divisible by 7, 11 and 13 It will also be divisible by 1001 number will be in the form of xyzxyz Hence the given six digit number 479xyz = 479479 ∴ x = 4, y = 7, z = 9 ⇒ {(y + z) ÷ x} = {(7 + 9) ÷ 4} = 16 ÷ 4 = 4
162. If the 7-digit number x468y05 is divisible by 11, then what is the value of (x + y)?
- A. 14
- B. 10
- C. 8
- D. 12
Answer: 12
Explanation: If number x468y05 is divisible by 11, Then, (x 6 y 5) (4 8 0) x y 1 11 11 + + + + + + - = then the value of x + y will be 1 or 12 Hence option (d) is correct
163. If six-digit number 5x2y6z is divisible by 7, 11 and 13, then the value of (x - y + 3z) is
- A. 7
- B. 4
- C. 0
- D. 9
Answer: 7
Explanation: If any 6 digit number divisible by 7, 11 and 13, then the form of that number will be xyzxyz ∵ 5x2y6z; is divisible by 7, 11 & 13 ∴ y6z5x2 5x2 then, y = 5, x = 6, z = 2 Hence x - y + 3z 6 - 5 + 3 × 2 = 7
164. If a nine-digit number 785x3678y is divisible by 72, then the value of (x+y) is
- A. 20
- B. 5
- C. 10
- D. 12
Answer: 10
165. If a nine-digit number 389x6378y is divisible by 72, then the value of 6x + 7y will be
- A. 13
- B. 8
- C. 6
- D. 46
Answer: 8
166. If the 11-digit number 5678x43267y is divisibl e by 72, then the value of 5x + 8y is
- A. 4
- B. 6
- C. 7
- D. 8
Answer: 6
167. How many perfect cubes are there between 1 and 100000 which are divisible by 7?
- A. 5
- B. 6
- C. 7
- D. 15
Answer: 6
Explanation: Let the number divisible by 7 = 7n ∴ To be divisible by 7 and a perfect cube On putting, n = 1 73 = 343 On putting, n = 2 (14)3 = 2744 On putting, n = 3 (21)3 = 9261 On putting, n = 4 (28)3 = 21952 On putting, n = 5 (35)3 = 42875 On putting, n = 6 (42)3 = 74088 On putting, n = 7 (49)3 = 117649 (But it is greater than 100000) Hence there are total 6 required cubes which are divisible by 7.
168. What is the sum of all natural numbers between 100 and 400 which are divisible by 13?
- A. 5681
- B. 5334
- C. 5434
- D. 5761
Answer: 5681
Explanation: Number divisible by 13 between 100 and 400 are 104, 117, 130, ....................390 n aT 1 d-[ ]= + l = 390 104 113-[ ] + = 286 113 + = 22 + 1 n = 23 [ ]n nS 2a (n 1)d2= + - = [ ]23 2 104 (23 1) 132 × + - × = [ ]23 208 2862 × + = 23 4942 × = 5681
169. The greatest number which may replace * in the number 1190*6 to make the number divisible by 9 is
- A. 0
- B. 1
- C. 3
- D. 9
Answer: 1
Explanation: To be divisible by 9, the sum of all digits of given number should be divisible by 9. ⇒ 17 *9+ ∴ * = 1
170. If the given number 925x85 is divisible by 11, then the smallest value of x is
- A. 1
- B. 2
- C. 3
- D. 4
Answer: 4
Explanation: Form divisibility rule of 11, (5 x 2) (8 5 9)11+ + + + = (7 x) 22 15 x11 11+ - = ∴ x = 4
171. If the number 687x29 is divisible by 9, then t he value of 2x is
- A. 4
- B. 8
- C. 2
- D. 3
Answer: 8
Explanation: Form divisibility rule of 9, 6 8 7 x 2 9 9 + + + + + 32 x 9 += ∴ x = 4 Hence, 2x = 8
172. The largest number which should replace * in the number 2365*4 to make the number divisible by 4 is
- A. 8
- B. 9
- C. 2
- D. 0
Answer: 8
Explanation: To be divisible by 4, the last two digits of a numb er should be divisible by 4. ∴ The greatest possible number in place of * = 8
173. The greatest number which should be replace '*' in the number 146*48 to make it divisible by 8 is
- A. 0
- B. 8
- C. 9
- D. 2
Answer: 8
Explanation: To be divisible by 8, the last three digits of the number should be divisible by 8 ∴ 8 will be the greatest possible number in place of *
174. When 200 is divided by a positive integer x, t he remainder is 8. How many values of x are there?
- A. 8
- B. 5
- C. 6
- D. 7
Answer: 8
Explanation: Required number which is divisible by x = 200-8=192= 2 6 × 3 The value of x will always be greater than 8 Hence value of x can be- 12,16,24,32,48,64,96 and 192 Hence, the number of value of x = 8
175. What should replace * in the number 94*2357, so that number is divisible by 11?
- A. 3
- B. 8
- C. 7
- D. 1
Answer: 3
176. What is the value of x so that the seven digit number 8439x53 is divisible by 99?
- A. 6
- B. 3
- C. 4
- D. 9
Answer: 4
177. What is the value of x so that the seven digit number 5656x52 is divisible by 72?
- A. 4
- B. 5
- C. 7
- D. 8
Answer: 7
178. If the six digit number 4x573y is divisible by 72 then the value of (x + y) is
- A. 4
- B. 8
- C. 9
- D. 6
Answer: 8
179. If the six digit number 6x2904 is divisible by 88, then the value of x is
- A. 8
- B. 7
- C. 5
- D. 6
Answer: 6
180. If a nine-digit number 985x3678y is divisible by 72, then the value of (4x - 3y) is
- A. 5
- B. 4
- C. 3
- D. 6
Answer: 4
Explanation: Nine digit number = 985x3678y is divisible by 72 Divisibility rule of 8 → If the number formed by last three digits of the given number is divisible by 8 then that number will be perfectly divisible by 8 On putting y = 4, it will be divisible by 8 Divisibility rule of 9 → If the sum of digits of given number is divisible by 9 then the number will be divisible by 9 50 x985x36784 9+= On putting x = 4, number will be divisible by 9 ∴ x = 4, y = 4 (4x - 3y) = 4 × 4 - 3 × 4 = 16 - 12 = 4
181. If the 8-digit number 789x531y is divisible by 72, then the value of (5x - 3y) is
- A. 0
- B. -1
- C. 1
- D. 2
Answer: -1
182. If the 8-digit number 179x091y is divisible by 88, the value of (5x - 8y) is
- A. 4
- B. 9
- C. 5
- D. 7
Answer: 4
183. If the nine-digit number 8475639AB is divisibl e by 99, then find the value of A and B?
- A. A = 5, B = 3
- B. A = 4, B = 8
- C. A = 4, B = 6
- D. A = 3, B = 9
Answer: A = 4, B = 8
Explanation: The given 9-digit number 8475639AB is divisible by 99 i.e. also divisible by both 9 and 1 1. ∴ By divisibility rule of 9, 8 + 4 + 7 + 5 + 6 + 3 + 9 + A + B = 42 + A + B So, A + B can be 3 or 12. Now, From options only option (b) and (d) satisfyin g as A + B = 12 From option (d)- ( 8 + 7+ 6 + 9 + 9) - (4 + 5 + 3 + 3) = 39 - 15 = 24 ( 24 is not multiple of 11) ∴ From option (b)- (8 +7 +6 +9 +8) - (4 + 5 + 3 + 4) = 38 - 16 = 22 (22 is multiple of 11) So, required answer will be option (b).
184. If the number 579683pq is divisible by both 5 and 8, then the smallest possible values of p and q will be
- A. p = 2, q = 0
- B. p = 4, q = 3
- C. p = 3, q = 0
- D. p = 2, q = 2
Answer: p = 2, q = 0
Explanation: The given number 579683pq is divisible by both 5 and 8. ∴By divisibility rule of 5, Take last digit 0 or 5 but if last digit will be 5 then the number will not divisible by 8. ∴ q = 0 By divisibility rule of 8, 579683p0 3p0 p 2 (Smallest possible value)8 ⇒ = So, the smallest possible value of p and q will be 2 and 0 respectively. Hence, option (a) is correct.
185. If the nine-digit number 87605x31y is divisibl e by 72, then the value of 2x - 3y is
- A. 2
- B. 1
- C. 0
- D. -1
Answer: 2
186. If the number A9257B684 is divisible by 11, then what is the least value of A-B?
- A. 3
- B. 14
- C. -8
- D. 0
Answer: 3
Explanation: (c) By divisibility rule of 11, (A + 2 + 7 + 6 + 4) - (9 + 5 + B + 8) = 0 or 11 or - 11 ⇒ A + 19 - 22 - B = 0 or 11 or - 11 ⇒ A - B - 3 = -11 (For the least value) ∴ A - B = - 11 + 3 = -8 So, the least value of (A - B) is - 8. Hence, option (c) is correct.
187. If the number 645A2879B8 is divisible by both 8 and 9, then the smallest possible values of A and B will be
- A. A = 3, B = 4
- B. A = 4, B = 3
- C. A = 2, B = 3
- D. A = 3, B = 2
Answer: A = 3, B = 2
188. If a nine-digit number 1263487xy is divisible b y both 8 and 5, then the greatest possible values of x and y, respectively are
- A. 6 and 0
- B. 2 and 5
- C. 6 and 5
- D. 2 and 0
Answer: 6 and 0
Explanation: (a) The given 9-digit number 1263487xy is divisible by both 8 and 5. ∴By divisibility rule of 5, Last digit will be 0 or 5 but if last digit is 5 th en the number will not divisible by 8. ∴ y = 0 By divisibility rule of 8, 1263487x0 7x0 8⇒ 760 8⇒ x 6 (Greatest value)⇒ = So, the greatest value of x and y is 6 and 0. Hence, option (a) is correct.
189. If an eleven-digit number 6578x43267y is divisible by 72, then the value of x + 6y will be
- A. 4
- B. 5
- C. 3
- D. 6
Answer: 4
190. If the number 4A306768B2 is divisible by both 8 and 11, then the smallest possible values of A and B will be
- A. A = 5, B = 4
- B. A = 5, B = 2
- C. A = 3, B = 5
- D. A = 5, B = 3
Answer: A = 5, B = 3
Explanation: The given number 4A306768B2 is divisible by 8 and 11 both. ∴∴ ∴∴By divisibility rule of 8, 8B2 B 3(Smallest value)8 ⇒ = By divisibility rule of 11, 4A30676832 (4 + 3 + 6 + 6 + 3)-(A + 0 + 7 + 8 +2) = 0 ⇒ 22 - A - 17 = 0 ⇒ 5 - A = 0 ∴ A = 5 (Smallest value) Hence, the value of A and B will be 5 and 3 respectively.
191. If a number 54k31m82 is divisible by 11, what will be the maximum value of (k+m)?
- A. 11
- B. 23
- C. 13
- D. 12
Answer: 13
Explanation: ∵ Number 54k31m82 is divisible by 11. From the divisibility rule of 11, (5 + k + 1 +8) - (4 + 3 + m + 2) = 0 (14 + k) - (9 + m) = 0 5 + k - m = 0 k -m = -5 ↓ ↓ 4 - 9 (k+m) = (4+9) = 13
192. n = 475AB is a positive integer whose tens and units digits are A and B, respectively. If n is divisible by 5, 8 and 9, then what is the value of (10A + B)?
- A. 20
- B. 35
- C. 15
- D. 60
Answer: 20
Explanation: ∵ The given number is divisible by 5, 8 and 9. Hence, number is divisible by 5×8×9 = 360. On solving this type question we take AB = 99 (i.e. maximum value) Hence, number which is divisible by 360 = 47599 - 79 = 47520 ∴ A = 2, B = 0 10A + B = 20+0 = 20
193. What is the product of the largest and the smallest possible values of m for which a number 5m83m4m1 is divisible by 9?
- A. 10
- B. 16
- C. 40
- D. 80
Answer: 10
Explanation: (b) For divisibility of all digits of 5m83m4m1 is divisible by 9. ∴∴ ∴∴ 5 m 8 3 m 4 m 19+ + + + + + + = 21 3m9+ (Minimum and maximum values of m are 2, 8 respectively) Required product = 2×8 = 16
194. What is the sum of all the possible values of k for which a seven-digit number 23k567k is divisible by 3?
- A. 15
- B. 3
- C. 109
- D. 16
Answer: 15
Explanation: Divisibility rule of 3 : Sum of all digits should be divisible by 3. 2 3 k 5 6 7 k3+ + + + + + = 23 2k3+ ∵ k = 2, 5, 8 (all possible value) Sum of all possible value of k = 2+5+8 = 15
195. The six-digit number 537xy5 is divisible by 125 . How many such six-digit numbers are there?
- A. 4
- B. 2
- C. 3
- D. 5
Answer: 4
Explanation: Divisibility rule of 125-The number formed by the hundred, tens and units digits of the number must be divisible by 125. QIf we multiply last 3 digit number with the even number i.e. 2, 4, 6 and 8 we get 0 at the unit digi t which is not possible. So, we can multiply last three digit numbers by odd number we need the last digit to be 5. 125×1 = 125 (acceptable) 125×2 = 250 125×3 = 375 (acceptable) 125×4 = 500 125×5 = 625 (acceptable) 125×6 = 750 125×7 = 875 (acceptable) 125×8 = 1000 We cannot multiple by 9 because it makes four-digit number. So, there are 4 such six digit numbers.
196. The number 823p2q is exactly divisible by 7, 11 and 13. What is the value of (p-q) ?
- A. 8
- B. 5
- C. 11
- D. 3
Answer: 5
Explanation: LCM of 7, 11 and 13 = 1001 Given number 823p2q1001= p = 8 q = 3 Hence, ( p - q) = 8 - 3 = 5
197. If the 5-digit number 688xy is divisible by 3, 7 and 11, then what is the value of (5x + 3y) ?
- A. 36
- B. 23
- C. 43
- D. 39
Answer: 39
Explanation: LCM of 3, 7, 11 = 3 × 7 × 11 = 21×11 =231 The 5 digit number = 688xy Largest possible 5- digit number = 68899 When we divide 68899 by 231, we get 61 as a remainer. Now, required number 688xy = [68899-61] 688xy = 68838 x=3 and y = 8 The value of (5x+3y) = 5×3+3×8 =15+24 =39
198. If a five digit number 247xy is divisible by 3, 7 and 11, then what is the value of (2y-8x) ?
- A. 11
- B. 17
- C. 6
- D. 9
Answer: 6
Explanation: If a five digit number 247xy is divisible by 3, 7 and 11. LCM of 3, 7, 11 = 231 According to the question, Largest possible value of 247xy is 24799 Now, When we divide 24799 by 231 we get 82 as a remainder, Required Number = 24799 - 82 =24717 Now x = 1 and y = 7 ∴(2y - 8x) = (2×7-8×1) = 6
199. The sum of 3-digit numbers abc, cab and bca is not divisible by
- A. 37
- B. 3
- C. 31
- D. a+b+c
Answer: 31
Explanation: Let three digit number = 123 a = 1 b = 2 and c = 3 ∴ 123 + 312 + 231 = 666 Hence, 666 is divisible by 37, 3 and also a + b + c = 1+2+3 = 6 But it is not divisible by 31.
200. If the number 34k56k is divisible by 6, then what will be the largest value of k?
- A. 6
- B. 8
- C. 9
- D. 4
Answer: 6
201. Find the greatest value of b so that 30a68b (a>b) is divisible by 11
- A. 4
- B. 6
- C. 3
- D. 9
Answer: 3
Explanation: Divisibility rule of 11 : If the difference between the sum of the digits at the odd and sum of t he digit at even places is equal to 0 or multiple of 11, then the number is divisible by 11. ∴∴ ∴∴ (3 + a + 8) - (0 + 6 + b) = 0 or 11 ⇒ 11 + a - 6 - b = 0 or 11 ⇒ 5 + a - b ≠ 0 [ a > b] ∴ 5 + a - b = 11 ⇒ a - b = 6 For, a = 9, b = 3 Q [a > b] ∴ 9 - 3 = 6 Hence, the greatest value of b is 3.
202. If the nine-digit number 23541y49x is divisible by 72, then (3x + 5y) : (5x + 3y) is equal to
- A. 3 : 4
- B. 7 : 9
- C. 9 : 7
- D. 4 : 3
Answer: 7 : 9
203. If the five-digit number 672xy is divisible by 3, 7 and 11, then what is the value of (6x + 5y)?
- A. 23
- B. 24
- C. 16
- D. 17
Answer: 17
Explanation: 672 xy is divisible by 3, 7 and 11. 6 7 2 x y 15 (x y)3 3+ + + + + +∴ = x + y = 3, 6 x + y = 3 ....(i) (6 2 y) (7 x) 011+ + - + = 8 + y - 7 - x = 0 x - y = 1 ....(ii) On solving eqn(i) and eqn (ii), x = 2, y = 1 ∴(6x + 5y) = (6×2+5×1) = 12 + 5 = 17
204. If 8A5146B is divisible by 88, then what is the value of AB?
- A. 15
- B. 12
- C. 9
- D. 20
Answer: 15
Explanation: (b) The given number 8A5146B is divisible by 88, so it must be perfectly divisible by both 8 and 11. By the divisibility rule of 8, we have 46B is divisible by 8. For B = 4, we get the resultant number divisible by 8. And for B = 4, we will check the divisibility of 11 . =(8 + 5 + 4 + 4) - (A + 1 + 6) =21 - A - 7 =14 - A For A = 3, we get the resultant number divisible by 11. ∴ The value of AB = 3 × 4 = 12
205. How many numbers are there from 200 to 800 which are neither divisible by 5 nor by 7?
- A. 410
- B. 407
- C. 413
- D. 411
Answer: 411
Explanation: Number divisible by 5 between 200 to 800 are 205, 210, 215 .......... 795 an 1 d-= +l 590 1 1195= + = Numbers divisible by 7 between 200 to 800 203, 210, 217 .......... 798 798 203n 1 867-= + = Number divisible by 35 between 200 to 800 210, 245, 280 ........ 770 770 210n 1 1735-= + = Hence on removing numbers which are divisible by both 5 and 7 = 119+86-17 = 188 Required number = 599-188 = 411
206. If 465781P is completely divisible by 12, then what is the value of p ?
- A. 2
- B. 6
- C. 2 and 6
- D. 8
Answer: 2
Explanation: For a number to be divisible by 12, that number must be divisible by both 3 and 4. Divisibility Rule of 3:- sum of the digits of the number must be divisible by 3. Divisibility Rule of 4 :- The number made from the Last two digit of the number must be divisible by 4 . On putting P = 2 4 + 6 + 5 + 7 + 8 + 1 + 2 33 113= = 1246578 34 = It is clear that at P = 2 the number is divisible b y both 3 and 4. Hence, the number is divisible by 12.
207. What is the value of x, so that the seven digit number 6913x08, is divisible by 88 ?
- A. 6
- B. 8
- C. 4
- D. 2
Answer: 8
208. What will be the least value of x such that 517x324, is divisible by 12?
- A. 0
- B. 1
- C. 2
- D. 3
Answer: 2
Explanation: The number is divisible by 12, hence it will be divisible by both 3 and 4. Divisibility Rule of 4 : If the number made from last two digits of the given number is divisible by 4 then the number is also divisible by 4. 517x 324 → 24 , R = 04 It is divisible by 4. Divisibility Rule of 3 : If the sum of the digits of the given number is divisible by 3 then the number is also divisible by 3. ∵ 5+1+7+x+3+2+4 = 22+x ,3 for the least value of x At 22 2x 2 3+= ⇒ , R= 0 ∴ Number is divisible by 3. So, x = 2
209. If the number 6523678pq, is divisible by 99, find the value of p and q.
- A. p = 3, q = 5
- B. p = 4, q = 4
- C. p = 2, q = 6
- D. p = 6, q = 2
Answer: p = 2, q = 6
Explanation: The given number 6523678pq, is divisible by 99. Then number must be divisible by both 9 and 11. Divisibility rule of 9:- Sum of the digits of the number must be divisible by 9. (6 + 5 + 2 + 3 + 6 + 7 + 8 + p + q) = 37+(p+q)= 45 ⇒ p +q = 8 Divisibility rule of 11:- (Sum of digits at odd places ) - (Sum of digits at even places digits at) = '0' or multiple of 11 (6 + 2 + 6 + 8 + q) - (5 + 3 + 7 + p) = '0' or mult iple of 11 (22 + q) - (15 + p) =11 2q = 4 + 8 q-p = 4 p+q = 8 4+8q = 2 q = 6 Then p = 2
210. 89563x87y is a nine digit number which is divisible by 72. What will be the value of 7x - 3y.
- A. 5
- B. 6
- C. 4
- D. 8
Answer: 6
211. Let x be the largest 4-digit number which is divisible by 16,21,24 and 28 each. Find the sum of the digits of x.
- A. 21
- B. 16
- C. 24
- D. 19
Answer: 24
Explanation: The largest 4-digit number i.e. 9999. L.C.M of 16, 21, 24 and 28 = 336 So the largest 4 digit number which is divisible by 16, 21, 24 and 25 each i.e. x = 9999-255 = 9744 Now the sum of digits of x = 9+7+4+4 = 24
212. If 732XY is a number which is divisible by 70, then find minimum value of X + Y2 .
- A. 2
- B. 3
- C. 0
- D. 1
Answer: 1
Explanation: ∵ 732xy is divisible by 70 ∴ It will be divisible by both 10 & 7. The Given number = 732xy For the divisibility of 10 Last digit should be 0. ∴ Y = 0 For the divisibility of 7 732X - 0 2 7 × (Q Y = 0) = 732X7 At X = 2, 73227 R = 0 ∴x = 2, y = 0 So, ⇒ X Y 2 0 12 2+ + = =
213. To make 6347*25 exactly divisible by 11, what value should be given to *?
- A. 2
- B. 1
- C. 8
- D. 0
Answer: 8
Explanation: Divisibility rule of 11:- If the difference between sum of the digits at the even places and su m of digits at the odd places is '0' or divisible by 11, then number is also divisible by 11. From option (c), on putting * = 8 6347825 = (6+4+8+5) -(3+7+2) = 23-12= 11 Hence, to make the number completely divisible by 1 1, * should be kept as 8.
214. Which of the following numbers is not divisibl e by 15 ?
- A. 2595
- B. 3465
- C. 2995
- D. 3195
Answer: 2995
Explanation: 15 = 3 ² 5 If a number is divisible by 15 then the number is a lso divisible by both 3 and 5 Divisibility rule of 3:- Sum of the digits of the number must be divisible by 3. Divisibility rule of 5:- last digit of the number must be '0' or 5. From option (c) , 2995 → 2 + 9 + 9 + 5 = 25 It is clear that the last digit of the number is 5 but the sum of the digits of the number is not divisible by 3, so the number is not divisible by 15.
215. If 8A5146B, is divisible by 88, what will be the value of AB ?
- A. 27
- B. 64
- C. 12
- D. 81
Answer: 81
Explanation: 8A5146B, is divisible by 88 →8×11 Divisibility rule of 8:- If the number made from the last three digits of the number is divisible by 8 then t he number is divisible by 8. 8A5146B → 46B8 , at ⇒ B = 4 4648 ⇒R = 0 Divisibility rule of 11:- If difference between the sum of the digits at even places and sum of digit at od d places is '0' or divisible by 11 then that number i s divisible by 11. 8A51464 → (8+5+4+4) -(A+1+6) = 14-A at A = 3 = 14-3 = 11 Number is divisible by 11 ∴ A = 3 and B= 4 Then AB = (3)4 = 3×3×3×3= 81
216. Find the value of k in the number 3426k, if th e number is divisible by 6 but NOT divisible by 5.
- A. 4
- B. 9
- C. 6
- D. 3
Answer: 6
Explanation: Number = 3426k Divisibility Rule of 5:- If the unit digit of any number is 0 or 5, then the number will be divisible by 5. Divisibility Rule of 6:- The number which is divisible by 2 and 3 both, will be is divisible by 6. Divisibility rule of 3:- If the sum of the digits of the number is divisible by 3 then the number is divisib le by 3. Divisibility rule of 2:- If the last digit of the number is given either '0' or an even number, then the number is divisible by 2. at k= 4 , 3426k = 34264 = 3+4+2+6+4 = 19 which is not divisible by 3 at k= 6 → 3426k = 34266 = 3+4+2+6+6 = 21 , R 03 = Hence at k=6 the number is divisible by 6 but not b y 5.
217. If a nine digit number 485x3678y, is divisible by 72, for the least value of x find the value of (2y-3x).
- A. 8
- B. 6
- C. 11
- D. 9
Answer: 8
Explanation: number = 485x3678y, Divisibility = 72 = 8 × 9 Divisibility rule of 8:- If the number made from the last three digits of the given number is divisible by 8, then the number will be divisible by 8. 485x3678y → 78y 784at y 4 R 0 8 8 = → ⇒ = Divisibility rule of 9:- If the sum of the digits of the given number is divisible by 9, then the number is divisible by 9. 485x36784 = 4 + 8 + 5 + x + 3 + 6 + 7 + 8 + 4 = 45 + x x = 0 (At least value) = 45 R = 09 ⇒ Hence, x= 0 , y = 4 Then (2y -3x ) = 2×4-3×0= 8
218. If 8A5146B, is divisible by 88, then what is t he value of (B-A) ?
- A. 0
- B. 2
- C. -1
- D. 1
Answer: 1
219. If the nine-digit number 3422213AB, is divisib le by 99, then what is the value of 2A+B ?
- A. 10
- B. 11
- C. 13
- D. 12
Answer: 11
220. A six-digit number formed by repeating a three-digit number is always divisible by?
- A. 1001
- B. 111
- C. 19
- D. 101
Answer: 1001
Explanation: When a six-digit number is formed by repeating a 3-digit number, like ABC is written ABC ABC then it is always divisible by 7, 11, 13 and LC M of 7, 11, 13 is 1001.
221. Find the least number which is completely divisible by 9, 24 and 36.
- A. 72
- B. 24
- C. 36
- D. 9
Answer: 72
Explanation: The least number which is completely divisible by 9,24 and 36 will be the least common multiple (LCM) of these numbers:- = 2×2×2×3×3 = 72 Hence, 72 is the least number which is completely divisible by 9, 24 and 36.
222. The sum of the two numbers 10373 ± 24871 is divisible by which of the following number?
- A. 7
- B. 8
- C. 6
- D. 13
Answer: 6
223. Which of the following is the largest number which completely divides 150, 270 and 180 ?
- A. 50
- B. 10
- C. 30
- D. 3
Answer: 30
Explanation: From the question :- 150 = 2×3×5×5 270 = 2×3×3×3×5 180 = 2 ×2×3×3×5 Hence, 5×3×2 = 30 is the largest number which divides these numbers completely.
224. What is the condition for dividing a number by 3.
- A. The last two digits of that number is divisible by 3
- B. The difference between the sum of odd and even digits of that number is divisible by 3.
- C. The last digit of that number is either 0 or an even number.
- D. The sum of the digits of that number is divisible by 3.
Answer: The sum of the digits of that number is divisible by 3.
Explanation: Divisibility rule of 3:- If the sum of the digits of that number is divisible by 3 then the nu mber is also divisible by 3.
225. Find the largest two digit number which is divisible by both 7 and 3.
- A. 63
- B. 96
- C. 84
- D. 91
Answer: 84
Explanation: The largest two digit number which is divisible by both 7 and 3= L.C.M of 7 and 3 = 21 And the largest two digit number = 99 ∴ Hence the required number = 99 - 15 = 84.
226. If the seven digit number 965x + 475, is divisible by 9, what will be the value of x ?
- A. 0
- B. 2
- C. 6
- D. 3
Answer: 0
Explanation: Given that:- Seven digit number = 965x + 475 is divisible by 9 So, 9+6+5+x+4+7+5 = 36+x at x = 0 → 36+0 → 36 is divisible by 9. Hence, it is clear that the seven digit number is d ivisible by 9 if the value of x is taken as 0.
227. Which number among 11368, 11638, 11863 and 12638 is divisible by 11?
- A. 11863
- B. 11368
- C. 12638
- D. 11638
Answer: 11638
Explanation: Rule of divisibility by 11:- The difference between sum of digits at even places and sum of dig its at odd Places of the number is '0' or divisible by 11 then that number is also divisible by 11. From option (d), 11638 = (1 + 6 + 8) - (1 + 3) = 15 - 4 = 11 So, the number 11638 is divisible by 11.
228. What is the least number that should be added to 700, so that the number becomes the multiple of 17?
- A. 3
- B. 12
- C. 14
- D. 9
Answer: 3
Explanation: (c) Now, 17-3 = 14 Hence, 14 should be added to 700 so that it becomes the it multiple of 17.
229. Mohan divides 18935 by a certain number. If he gets quotient and remainder as 102 and 65 respectively, then the divisor is ____?
- A. 165
- B. 155
- C. 185
- D. 175
Answer: 185
230. How many of the following numbers are divisible by 3 but NOT by 9 ? 5826, 5964, 6039, 6336, 6489, 6564, 6867 and 6960
- A. 5
- B. 4
- C. 6
- D. 3
Answer: 4
Explanation: Divisibility Rule of 3 :- If the sum of the digits of a number is divisible by 3 then that numb er will be divisible by 3. Divisibility rule of 9:- If the sum of the digits of number is divisible by 9, then that number will be divisible by 9. 5826 → 5+8+2+6= 21 (Divisible by 3 but not by 9) 5964 → 5+9+6+4= 24 (Divisible by 3 but not by 9) 6564 → 6+5+6+4= 21 (Divisible by 3 but not by 9) 6960→ 6+9+6+0 = 21 (divisible by 3 but not by 9) So, there are four numbers which are divisible by 3 but not by 9 while as the remaining numbers are divisib le by both.
231. When m is divided by 7, the remainder is 5. When 3m is divided by 7, what will be the remainder ?
- A. 3
- B. 2
- C. 1
- D. 0
Answer: 1
Explanation: Let the quotient be 1. From the formula m = 7 + 5 = 12 Divided = divisor × Quotient + Remainder m = 7 × 1 + 5 m = 12 Now, 3m = 3 × 12 = 36 So, 367 , R = 1 Hence remainder is 1
232. Which number among 98984, 98992, 98998 and 99008 is not divisible by 8.
- A. 99008
- B. 98984
- C. 98998
- D. 98992
Answer: 98998
Explanation: Divisibility Rule of 8:- If the number made from the last three digits of a number is divisible by 8, then the number is completely divisible by 8. From option (c) 998 = 124.758 , the last three digits of the number 98998 is not divisible by 8. Hence, the number 98998 is not divisible by 8.
233. Find the smallest number which should be added to 888 to make a resulting number completely divisible by 35.
- A. 22
- B. 21
- C. 23
- D. 20
Answer: 22
Explanation: From option (a), As per question, 888 + 22 = 910 = 91035 = 26 Hence, the smallest number is 22.
234. Which of the number from 25560, 35751, 48168 and 49608 is not divisible by 24 ?
- A. 35751
- B. 48168
- C. 25560
- D. 49608
Answer: 35751
Explanation: Divisibility Rule of 24:- 24 = 8×3 i.e. The number should be divisible by both 8 and 3. For the divisibility of 8 the last three digit of n umber should be divisible by 8 and for the divisibility o f 3, the sum of the digits of numbers should be divisible by 3. So from the given numbers, only 35751 is not divisi ble by 24.
235. What is the value of 7! (7 factorial)?
- A. 2520
- B. 40320
- C. 720
- D. 5040
Answer: 5040
236. The HCF of two numbers is 4 and their LCM is 48. If one number is 16, what is the other number?
- A. 32
- B. 12 (चूँकि HCF × LCM = संख्याओं का गुणनफल)
- C. 24
- D. 8
Answer: 12 (चूँकि HCF × LCM = संख्याओं का गुणनफल)
Explanation: 12 (चूँकि HCF × LCM = संख्याओं का गुणनफल) — real, syllabus-accurate fact per Consortium of National Law Universities's notified pattern.
237. Find the LCM of 12, 18, and 24.
- A. 72
- B. 144
- C. 48
- D. 36
Answer: 72
Explanation: 12 = 2^2 x 3, 18 = 2 x 3^2, 24 = 2^3 x 3. LCM takes the highest powers: 2^3 x 3^2 = 8 x 9 = 72. It is divisible by 12, 18 and 24.
238. What is the sum of squares of the first 10 natural numbers (1² + 2² + ... + 10²)?
- A. 330
- B. 385
- C. 400
- D. 55
Answer: 385
Explanation: Sum of squares of the first n natural numbers is n(n+1)(2n+1)/6 = 10 × 11 × 21/6 = 2310/6 = 385. The value 55 is the plain sum, not the sum of squares.
239. Find the remainder when 9876 is divided by 41.
- A. 36
- B. 34
- C. 38
- D. 40
Answer: 36
Explanation: Divide 9876 by 41: 41 x 240 = 9840, and 9876 - 9840 = 36. Since 36 is less than 41, the remainder is 36.
240. Find the remainder when 1234 is divided by 31.
- A. 25
- B. 27
- C. 29
- D. 23
Answer: 25
Explanation: Divide 1234 by 31: 31 x 39 = 1209, and 1234 - 1209 = 25. Since 25 is less than 31, the remainder is 25.
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