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321. Express 0.875 as a fraction in lowest terms.
- A. 7/8
- B. 3/4
- C. 5/8
- D. 7/9
Answer: 7/8
Explanation: 0.875 = 875/1,000. The HCF of 875 and 1,000 is 125, so dividing both by 125 gives 7/8 in lowest terms. The other options are either not equal in value or not fully reduced.
322. Find the remainder when 2000 is divided by 17.
- A. 11
- B. 7
- C. 13
- D. 9
Answer: 11
Explanation: Divide 2000 by 17: 17 x 117 = 1989, and 2000 - 1989 = 11. Since 11 is less than 17, the remainder is 11.
323. Find the LCM of 18, 24 and 30.
- A. 720
- B. 180
- C. 360
- D. 270
Answer: 360
Explanation: 18 = 2 x 3^2, 24 = 2^3 x 3 and 30 = 2 x 3 x 5. Taking the highest power of each prime: 2^3 x 3^2 x 5 = 8 x 9 x 5 = 360.
324. Find the HCF of 144, 180 and 192.
- A. 36
- B. 6
- C. 12
- D. 24
Answer: 12
Explanation: 144 = 2^4 x 3^2, 180 = 2^2 x 3^2 x 5 and 192 = 2^6 x 3. The common factors with the lowest powers are 2^2 and 3, so HCF = 4 x 3 = 12.
325. Which of the following is NOT a prime number: 97, 101, 217, 199?
- A. 199
- B. 101
- C. 97
- D. 217
Answer: 217
Explanation: 217 = 7 x 31, so it is composite. 97, 101 and 199 are not divisible by any prime up to their square roots (2, 3, 5, 7, 11, 13), so they are prime.
326. Find the smallest 3-digit number exactly divisible by 19.
- A. 119
- B. 114
- C. 108
- D. 100
Answer: 114
Explanation: 19 x 5 = 95 is only 2-digit and 19 x 6 = 114 is the first multiple that reaches 100 or more. So the smallest 3-digit number divisible by 19 is 114.
327. Find the unit digit of 6^25.
- A. 1
- B. 0
- C. 5
- D. 6
Answer: 6
Explanation: The unit digit of 6^n depends only on the unit digit of the base, and 6 multiplied by itself always ends in 6. So 6^25 ends in 6.
328. Find the unit digit of 2^63.
- A. 4
- B. 2
- C. 6
- D. 8
Answer: 8
Explanation: The unit digits of powers of 2 repeat in a cycle of 4: 2, 4, 8, 6. Since 63 divided by 4 leaves remainder 3, we take position 3 in the cycle, giving unit digit 8.
329. Two numbers are in the ratio 5:6 and their LCM is 120. Find the numbers.
- A. 10 and 12
- B. 25 and 30
- C. 20 and 24
- D. 15 and 18
Answer: 20 and 24
Explanation: Let the numbers be 5k and 6k. Their LCM is 30k (since 5 and 6 are coprime), so 30k = 120 and k = 4. The numbers are 20 and 24, and LCM(20, 24) = 120 checks.
330. What is the smallest number that must be added to 758 to make it exactly divisible by 11?
- A. 2
- B. 10
- C. 9
- D. 1
Answer: 1
Explanation: 758 divided by 11 gives 11 x 68 = 748 with remainder 10. The next multiple is 11 x 69 = 759, so adding 759 - 758 = 1 makes 758 divisible by 11.
331. Find the LCM of 9, 15 and 20.
- A. 360
- B. 180
- C. 90
- D. 135
Answer: 180
Explanation: Factorise: 9 = 3^2, 15 = 3 x 5, 20 = 2^2 x 5. The LCM takes the highest power of each prime: 2^2 x 3^2 x 5 = 4 x 9 x 5 = 180.
332. Find the HCF of 105 and 180.
- A. 21
- B. 45
- C. 15
- D. 9
Answer: 15
Explanation: Factorise: 105 = 3 x 5 x 7 and 180 = 2^2 x 3^2 x 5. The HCF is the product of the common prime factors with the lowest powers, which is 15. It divides 105 exactly 7 times and 180 exactly 12 times.
333. Find the sum of squares of the first 10 natural numbers.
- A. 365
- B. 350
- C. 385
- D. 400
Answer: 385
Explanation: The sum of squares of the first n natural numbers is n(n+1)(2n+1)/6. For n = 10: 10 x 11 x 21 / 6 = 2310 / 6 = 385.
334. Find the sum of the first 30 natural numbers.
- A. 480
- B. 435
- C. 450
- D. 465
Answer: 465
Explanation: The sum of the first n natural numbers is n(n+1)/2. For n = 30 this is 30 x 31 / 2 = 930 / 2 = 465.
335. Express 0.45 as a fraction in lowest terms.
- A. 3/20
- B. 4/5
- C. 9/20
- D. 9/10
Answer: 9/20
Explanation: 0.45 = 45/100. The HCF of 45 and 100 is 5, so dividing both by 5 gives 9/20 in lowest terms. The other options are either not equal in value or not fully reduced.
336. Express 0.625 as a fraction in lowest terms.
- A. 5/8
- B. 7/8
- C. 3/8
- D. 5/9
Answer: 5/8
Explanation: 0.625 = 625/1,000. The HCF of 625 and 1,000 is 125, so dividing both by 125 gives 5/8 in lowest terms. The other options are either not equal in value or not fully reduced.
337. Find the largest 4-digit number exactly divisible by 27.
- A. 9981
- B. 9990
- C. 9963
- D. 9999
Answer: 9990
Explanation: The largest 4-digit number is 9999. 9999 divided by 27 leaves remainder 9999 - 27 x 370 = 9999 - 9990 = 9, so subtracting 9 gives 9990 = 27 x 370.
338. Find the smallest 5-digit number exactly divisible by 23.
- A. 10000
- B. 10012
- C. 10005
- D. 10023
Answer: 10005
Explanation: The smallest 5-digit number is 10000. 10000 divided by 23 leaves remainder 10000 - 23 x 434 = 10000 - 9982 = 18, so add 23 - 18 = 5 to get 10005 = 23 x 435.
339. Find the unit digit of 9^100.
- A. 1
- B. 9
- C. 7
- D. 3
Answer: 1
Explanation: The unit digits of powers of 9 repeat in a cycle of 2: 9, 1. Since 100 divided by 2 leaves remainder 0, we take position 2 in the cycle, giving unit digit 1.
340. Find the unit digit of 3^41.
- A. 7
- B. 3
- C. 9
- D. 1
Answer: 3
Explanation: The unit digits of powers of 3 repeat in a cycle of 4: 3, 9, 7, 1. Since 41 divided by 4 leaves remainder 1, we take position 1 in the cycle, giving unit digit 3.
341. Find the LCM of 24 and 36.
- A. 72
- B. 144
- C. 36
- D. 108
Answer: 72
Explanation: HCF of 24 and 36 is 12. Using LCM = (product) / HCF = 24 x 36 / 12 = 864 / 12 = 72. Check: 72 divides evenly by both 24 and 36.
342. Find the HCF of 84 and 126.
- A. 14
- B. 21
- C. 63
- D. 42
Answer: 42
Explanation: Factorise: 84 = 2^2 x 3 x 7 and 126 = 2 x 3^2 x 7. The HCF is the product of the common prime factors with the lowest powers, which is 42. It divides 84 exactly 2 times and 126 exactly 3 times.
343. Find the smallest number that must be added to 1000 to make it exactly divisible by 19.
- A. 5
- B. 12
- C. 9
- D. 7
Answer: 7
Explanation: 1000 divided by 19 gives 19 x 52 = 988 with remainder 12. The next multiple is 19 x 53 = 1007, so the number to add is 1007 - 1000 = 7 (equivalently 19 - 12).
344. Express 0.125 as a fraction in lowest terms.
- A. 1/9
- B. 1/8
- C. 1/4
- D. 1/5
Answer: 1/8
Explanation: 0.125 = 125/1,000. The HCF of 125 and 1,000 is 125, so dividing both by 125 gives 1/8 in lowest terms. The other options are either not equal in value or not fully reduced.
345. Find the sum of the first 20 natural numbers.
- A. 220
- B. 200
- C. 190
- D. 210
Answer: 210
Explanation: The sum of the first n natural numbers is n(n+1)/2. For n = 20 this is 20 x 21 / 2 = 420 / 2 = 210.
346. Find the largest 3-digit number divisible by 17.
- A. 991
- B. 983
- C. 969
- D. 986
Answer: 986
Explanation: The largest 3-digit number is 999. 999 divided by 17 leaves remainder 999 - 17 x 58 = 999 - 986 = 13, so subtracting 13 gives 986 = 17 x 58.
347. Find the unit digit of 8^57.
- A. 6
- B. 8
- C. 4
- D. 2
Answer: 8
Explanation: The unit digits of powers of 8 repeat in a cycle of 4: 8, 4, 2, 6. Since 57 divided by 4 leaves remainder 1, we take position 1 in the cycle, giving unit digit 8.
348. Find the LCM of 15 and 20.
- A. 100
- B. 60
- C. 30
- D. 45
Answer: 60
Explanation: LCM = (product of the numbers) / HCF. HCF of 15 and 20 is 5, so LCM = 15 x 20 / 5 = 300 / 5 = 60. Hence 60 is the smallest number that both 15 and 20 divide exactly.
349. Find the HCF of 48 and 60.
- A. 6
- B. 15
- C. 24
- D. 12
Answer: 12
Explanation: Factorise: 48 = 2^4 x 3 and 60 = 2^2 x 3 x 5. The HCF is the product of the common prime factors with the lowest powers, which is 12. It divides 48 exactly 4 times and 60 exactly 5 times.
350. Two numbers are in ratio 3:4 and their LCM is 84. Find the numbers.
- A. 18 and 24
- B. 21 and 28
- C. 15 and 20
- D. 12 and 16
Answer: 21 and 28
Explanation: Let the numbers be 3k and 4k. Since 3 and 4 are coprime, their LCM is 12k, so 12k = 84 and k = 7. The numbers are 21 and 28, and LCM(21, 28) = 84 checks.
351. Find the smallest 4-digit number exactly divisible by 12.
- A. 1000
- B. 1012
- C. 1008
- D. 996
Answer: 1008
Explanation: The smallest 4-digit number is 1000. 1000 divided by 12 leaves remainder 1000 - 12 x 83 = 1000 - 996 = 4, so add 12 - 4 = 8 to get 1008 = 12 x 84.
352. Express 0.375 as a fraction in lowest terms.
- A. 3/7
- B. 5/8
- C. 3/10
- D. 3/8
Answer: 3/8
Explanation: 0.375 = 375/1,000. The HCF of 375 and 1,000 is 125, so dividing both by 125 gives 3/8 in lowest terms. The other options are either not equal in value or not fully reduced.
353. What is the unit digit of 7^95?
- A. 7
- B. 9
- C. 1
- D. 3
Answer: 3
Explanation: The unit digits of powers of 7 repeat as 7, 9, 3, 1 (cycle of 4). 95 divided by 4 leaves remainder 3, so 7^95 has the third digit of the cycle, which is 3.
354. Which of the following is a prime number: 91, 97, 111, 119?
- A. 111
- B. 119
- C. 97
- D. 91
Answer: 97
Explanation: 91 = 7 x 13, 111 = 3 x 37 and 119 = 7 x 17 all have factors other than 1 and themselves. 97 is not divisible by 2, 3, 5 or 7 (the primes up to its square root), so it is prime.
355. What is the LCM of 12 and 18?
- A. 12
- B. 36
- C. 24
- D. 72
Answer: 36
Explanation: HCF of 12 and 18 is 6. Using LCM = (product) / HCF = 12 x 18 / 6 = 216 / 6 = 36. Check: 36 divides evenly by both 12 and 18.
356. What is the HCF of 36 and 60?
- A. 6
- B. 12
- C. 18
- D. 24
Answer: 12
Explanation: Factorise: 36 = 2^2 x 3^2 and 60 = 2^2 x 3 x 5. The HCF is the product of the common prime factors with the lowest powers, which is 12. It divides 36 exactly 3 times and 60 exactly 5 times.
357. Is 4356 divisible by 4?
- A. No
- B. Only divisible by 2, not 4
- C. Yes (last two digits 56 ÷ 4 = 14)
- D. Cannot be determined
Answer: Yes (last two digits 56 ÷ 4 = 14)
Explanation: A number is divisible by 4 if its last two digits are divisible by 4. Here the last two digits are 56, and 56 / 4 = 14 exactly. So 4356 / 4 = 1089, with no remainder.
358. What is the sum of all natural numbers from 1 to 100?
- A. 5050
- B. 6000
- C. 5000
- D. 5052
Answer: 5050
359. If the sum of two numbers is 14 and their difference is 10. Find the product of these two numbers.
- A. 24
- B. 22
- C. 20
- D. 18
Answer: 24
360. Consider the following statements for the sequence of numbers given below: 11, 111, 1111, 11111, …. 1. Each number can be expressed in the form (4m + 3), where m is a natural number. 2. Some numbers are squares. Which of the above statements is/are correct?
- A. 1 only
- B. 2 only
- C. Both 1 and 2
- D. neither 1 nor 2)
Answer: 1 only
Explanation: 11, 111, 1111, 11111, ….. Let m = 2 ⇒ 4 × 2 + 3 = 11 m = 27 ⇒ 4 × 27 + 3 = 111 Each number can be expressed in the form (4m + 3) where m is a natural number Hence, statement 1 is only correct.
361. What is the number of trees planted in the field in row and column? [IBPS—Bank Spl. Officer (Marketing) Exam, 2016] I. Number of columns is more than the number of rows by 4. II. Number of columns is 20.
- A. If the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question.
- B. If the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question.
- C. If the data either in Statement I alone or in Statement II alone are sufficient to answer the question.
- D. If the data given in both Statements I and II together are not sufficient to answer the question.
Answer: If the data given in both Statements I and II together are not sufficient to answer the question.
Explanation: The data in both the statements I and II are not sufficient to answer the question.
362. Two consecutive even positive integers, sum of the squares of which is 1060, are
- A. 12 and 14
- B. 20 and 22
- C. 22 and 24
- D. 15 and 18
Answer: 22 and 24
363. The number of three digit numbers which are multiples of 9 are
- A. 100
- B. 99
- C. 98
- D. 101
Answer: 100
Explanation: The first 3-digit number which is divisible by 9 is 108 and last three digit number which is divisible by 9 is 999. So, we have an AP with a = 108, d = 9 and an = 999 ∴ an = a + (n – 1)d ⇒ 999 = 108 + (n – 1) 9 ⇒ 999 – 108 = (n – 1) 9 ⇒ 891 = (n – 1) 9 ⇒ (n – 1) = 891 9 = 99 ⇒ n + 99 + 1 = 100
364. The least number of five digit is exactly divisible by 88 is
- A. 10032
- B. 10132
- C. 10088
- D. 10023
Answer: 10032
Explanation: A megabyte is 1.048, 576 bytes or 1,024 kilobytes. It conveniently expression the binary multiples inherent in digital computer memory architectures. However, megabyte architectures. However, megabyte is also taken to mean 1000 × 1024 (1024000) bytes.
365. How many prime numbers are there between 100 to 200?
- A. 21
- B. 20
- C. 16
- D. 11
Answer: 21
366. What minimum value should be assigned to *, so that 2361*48 is exactly divisible by 9?
- A. 2
- B. 3
- C. 9
- D. 4
Answer: 3
Explanation: 2361*48 will be divisible by 9 if the sum of the digits of the given number is divisible by 9. 2 + 3 + 6 + 1 + * + 4 + 8 i.e. (24 + *) is divisible by 9. Clearly, * = 3 because 27 is divisible by 9.
367. The difference between the greatest and the least four digit numbers that beings with 3 and ends with 5 is
- A. 900
- B. 909
- C. 999
- D. 990
Answer: 990
368. If (the place value of 5 in 15201) + (the place value of 6 in 2659) = 7 × ____, then the number of the blank space is:
- A. 800
- B. 80
- C. 90
- D. 900
Answer: 800
369. If (1001 × 111) = 110000 + (11 × ___), then the number in the blank space is
- A. 121
- B. 211
- C. 101
- D. 1111
Answer: 101
370. If a + b + c = 6 and ab + bc + ca = 10 then the value of a3 + b3 + c3 – 3abc is
- A. 36
- B. 48
- C. 42
- D. 40
Answer: 36
371. Among the following statements, the statement which is not correct is:
- A. Every natural number is an integer.
- B. Every natural number is a real number.
- C. Every real number is a rational number.
- D. Every integer is a rational number.
Answer: Every real number is a rational number.
Explanation: Every real number is a rational number is not a correct statement.
372. Which is not a prime number?
- A. 13
- B. 19
- C. 21
- D. 17
Answer: 21
Explanation: 21 = 3 × 7 is not a prime number because 21 is a composite number.
373. How many numbers will be there between 300 and 500, where 4 comes only one time?
- A. 89
- B. 99
- C. 110
- D. 120
Answer: 99
Explanation: From 300 to 399, we note that when ‘4’ comes only one time = 19 such instances. From 400 to 499, we note that when ‘4’ comes only one time = 80 such instances. So, total = (19 + 80) = 99 such instances
374. What is the number of digits in the number (1024)4 × (125)11?
- A. 35
- B. 36
- C. 37
- D. 38
Answer: 36
Explanation: (1024)4 × (125)11 = (210) 4 × (53)11 = 240 × 533 = 27 × (233 × 533) = 27 × 1033 = 128 × 1033. Clearly, the number has 1, 2, 8 and thirty-three zeros, i.e. (3 + 33) = 36 digits in all.
375. When 10025 – 25 is written in decimal notation, the sum of its digits is
- A. 444
- B. 445
- C. 446
- D. 448
Answer: 444
Explanation: 10025 – 25 = (102) 25 – 25 = 1050 – 25 = 50 zeros 48 times 1000 ..... 00 – 25 9999 ......... 9975 = ∴ Sum of digits = (48 × 9) + 7 + 5 = 432 + 7 + 5 = 444.
376. Find the unit's digit in the sum of the fifth powers of the first 100 natural numbers.
- A. 0
- B. 2
- C. 5
- D. 8
Answer: 0
Explanation: Let N = 15 + 25 + 35 + ...........+ (100)5. Then, N = (15 + 25 + 35 +.......+ 105) + (115 + 125 +...........+ 205) + (215 + 225 +..........+ 305) +......+ (915 + 925 + ...........+ 1005) = N1 + N2 + N3 +........+ N10. Since each one of N1, N2, N3, .........., N10 has the same unit's digit of its terms, so unit's digit of each one of N1, N2,......, N10 is also the same. ∴ Unit's digit in N = 10 × Unit's digit of N1 = 0.
377. If a2 + b2 + c2 = 1, what is the maximum value of abc?
- A. 1 3
- B. 1 33
- C. 2 3
- D. 1
Answer: 1 33
378. How many zeros will be required to number the pages of a book containing 1000 pages?
- A. 168
- B. 184
- C. 192
- D. 216
Answer: 192
Explanation: The pages of the book may be divided into 10 groups: (1 – 100), (101 – 200), (201 – 300),..........., (901 – 1000). Clearly, for the first group, one needs 11 zeros. For second to ninth groups, one needs 20 zeros each. For the tenth group, one needs 21 zeros. So, total number of zeros required = 11 + 8 × 20 + 21 = 192.
379. Which one of the following is the common factor of (4743 + 4343) and (4747 + 4347)?
- A. (47 – 43)
- B. (47 + 43)
- C. (4743 + 4343)
- D. None of these
Answer: (47 + 43)
Explanation: When n is odd, (xn + an) is always divisible by (x + a). ∴ Each one of (4743 + 4343) and (4747 + 4347) is divisible by (47 + 43).
380. What is the remainder when 1736 is divided by 36?
- A. 1
- B. 7
- C. 19
- D. 29
Answer: 1
Explanation: 1736 = (172)18 = (289)18. Now, [(289)18 – 1] is divisible by (289 – 1), i.e. 288 ⇒ (1736 – 1) is divisible by 288 and hence by 36 ⇒ On dividing 1736 by 36, we get 1 as remainder.
381. What is the remainder when 461 is divided by 51?
- A. 20
- B. 41
- C. 50
- D. None of these
Answer: None of these
Explanation: 461 = 4 × 460 = 4 × (44)15 = 4 × (256)15. Now, (xn – an) is divisible by (x – a) for all values of n. ∴ (25615 – 1) is divisible by (256 – 1) i.e. 255 and hence by 51. ⇒ On dividing (256)15 by 51, we get 1 as remainder ⇒ On dividing 460 by 51, we get 1 as remainder ⇒ On dividing 461 by 51, remainder obtained = (4 × 1) = 4.
382. Find the remainder when N is divided by 168.
- A. 33
- B. 67
- C. 129
- D. 153
Answer: 33
383. Find the last two digits of N.
- A. 00
- B. 13
- C. 19
- D. 23
Answer: 13
384. Let N = 553 + 173 – 723. Then, N is divisible by
- A. both 7 and 13
- B. both 3 and 13
- C. both 17 and 7
- D. both 3 and 17
Answer: both 3 and 17
Explanation: Let a = 55, b = 17. Then, N = a3 + b3 – (a + b) 3 = a3 + b3 – [a3 + b3 + 3ab (a + b)] = – 3ab (a + b) = – 3 × 55 × 17 × 72. Clearly, N is divisible by both 3 and 17.
385. n being any odd number greater than 1, n65 – n is always divisible by
- A. 5
- B. 13
- C. 24
- D. None of these
Answer: 24
Explanation: n65 – n = n (n64 – 1) = n(n32 – 1) (n32 + 1) = n (n16 – 1) (n16 + 1) (n32 + 1) = n (n8 – 1) (n8 + 1) (n16 + 1) (n32 + 1) = n (n4 – 1) (n4 + 1) (n8 + 1) (n16 + 1) (n32 + 1) = n (n2 – 1) (n2 + 1) (n4 + 1) (n8 + 1) (n16 + 1) (n32 + 1) = (n – 1) n (n + 1) (n2 + 1) (n4 + 1) (n8 + 1) (n16 + 1) (n32 + 1). Clearly, (n – 1), n and (n + 1) are three consecutive numbers and they have to be multiples of 2, 3 and 4 as n is odd. Thus, the given number is definitely a multiple of 24.
386. The number (248 – 1) is exactly divisible by two numbers between 60 and 70. The numbers are
- A. 63 and 65
- B. 63 and 67
- C. 61 and 65
- D. 65 and 67
Answer: 63 and 65
Explanation: (248 – 1) = [(26)8 – 1] = (64)8 – 1. When n is even, (xn – an) is completely divisible by both (x – a) and (x + a). ∴ (648 – 18) is divisible by both (64 – 1) and (64 + 1) ⇒ (248 – 1) is divisible by both 63 and 65.
387. It is given that (232 + 1) is exactly divisible by a certain number. Which of the following is also definitely divisible by the same number?
- A. 216 + 1
- B. 216 – 1
- C. 7 × 233
- D. 296 + 1
Answer: 296 + 1
Explanation: Let 232 = x. Then, (232 + 1) = (x + 1). Let (x + 1) be completely divisible by the natural number N. Then, (296 + 1) = [(232) 3 + 1] = (x3 + 1) = (x + 1) (x2 – x + 1), which is completely divisible by N since (x + 1) is divisible by N.
388. 76n – 66n, where n is an integer > 0, is divisible by
- A. 13
- B. 127
- C. 559
- D. All of these
Answer: All of these
Explanation: When n is even, (xn – an) is divisible by both (x – a) as well as (x + a). Now, (76n – 66n) = [(73)2n – (63)2n] = [(343)2n – (216)2n]. ∴ (76n – 66n) is divisible by both (7 – 6) and (7 + 6) ⇒ (76n – 66n) is divisible by 13. And, [(343)2n – (216)2n] is divisible by both (343 – 216) and (343 + 216) ⇒ (76n – 66n) is divisible by both 127 and 559.
389. When 2256 is divided by 17, the remainder would be
- A. 1
- B. 14
- C. 16
- D. None of these
Answer: 1
Explanation: When n is even, (xn – an) is divisible by (x + a). Now, 2256 = (24) 64 = (16)64. ∴ (1664 – 164) is divisible by (16 + 1) ⇒ (1664 – 1) is divisible by 17 ⇒ (2256 – 1) is divisible by 17 ⇒ On dividing 2256 by 17, we get 1 as remainder.
390. The remainder when (1523 + 2323) is divided by 19, is
- A. 0
- B. 4
- C. 15
- D. 18
Answer: 0
Explanation: (xn + an) is divisible by (x + a) when n is odd. ∴ (1523 + 2323) is divisible by (15 + 23) ⇒ (1523 + 2323) is divisible by 38 and hence by 19 ⇒ On dividing (1523 + 2323) by 19, we get 0 as remainder.
391. By how many of the following numbers is 212 – 1 divisible? 2, 3, 5, 7, 10, 11, 13, 14
- A. 4
- B. 5
- C. 6
- D. 7
Answer: 4
Explanation: (212 – 1) = (4096 – 1) = 4095, which is clearly divisible by 3, 5, 7 and 13 i.e. four numbers in all.
392. The remainder when 260 is divided by 5 equals
- A. 0
- B. 1
- C. 2
- D. 3
Answer: 1
Explanation: 260 = (22)30 = 430. When n is even, (xn – an) is divisible by (x + a). ∴ (430 – 130) is divisible by (4 + 1) ⇒ (430 – 1) is divisible by 5 ⇒ (260 – 1) is divisible by 5. ⇒ On dividing 260 by 5, we get 1 as remainder.
393. The remainder when 784 is divided by 342 is
- A. 0
- B. 1
- C. 49
- D. 341
Answer: 1
Explanation: 784 = (73)28 = (343)28. Now, (xn – an) is divisible by (x – a) for all values of n. ∴ [(343)28 – 1] is divisible by (343 – 1) ⇒ [(343)28 – 1] is divisible by 342 ⇒ (784 – 1) is divisible by 342 ⇒ On dividing 784 by 342, we get 1 as remainder.
394. One less than (49)15 is exactly divisible by
- A. 8
- B. 14
- C. 50
- D. 51
Answer: 8
Explanation: (4915 – 1) = (72)15 – 1 = 730 – 1. Now, when n is even, (xn – an) is divisible by both (x – a) and (x + a). ∴ (730 – 1) is divisible by both (7 – 1) and (7 + 1) i.e. by both 6 and 8. Thus, [(49)15 – 1] is divisible by both 6 and 8.
395. If (6767 + 67) is divided by 68, the remainder is
- A. 1
- B. 63
- C. 66
- D. 67
Answer: 66
Explanation: (xn + an) is divisible by (x + a) when n is odd. ∴ (6767 + 167) is divisible by (67 + 1) ⇒ (6767 + 1) is divisible by 68 ⇒ On dividing (6767 + 67) by 68, we get (67 – 1) = 66 as remainder.
396. 2525 is divided by 26, the remainder is
- A. 1
- B. 2
- C. 24
- D. 25
Answer: 25
Explanation: (xn + an) is divisible by (x + a) when n is odd. ∴ (2525 + 125) is divisible by (25 + 1) ⇒ (2525 + 1) is divisible by 26 ⇒ On dividing 2525 by 26, we get (26 – 1) = 25 as remainder.
397. If (12n + 1) is divisible by 13, then n is
- A. 1 only
- B. 12 only
- C. any odd integer
- D. any even integer
Answer: any odd integer
Explanation: (xn + an) is divisible by (x + a) when n is odd. ∴ (12n + 1) is divisible by (12 + 1) i.e. 13 when n is odd.
398. If n is even, (6n – 1) is divisible by
- A. 6
- B. 30
- C. 35
- D. 37
Answer: 35
Explanation: When n is even, (xn – an) is divisible by both (x – a) and (x + a). So, (6n – 1) is divisible by both (6 – 1) and (6 + 1) ⇒ (6n – 1) is divisible by both 5 and 7 ⇒ (6n – 1) is divisible by (5 × 7), i.e. 35. [∵ 5 and 7 are co-primes]
399. (96 + 1) when divided by 8, would leave a remainder of
- A. 0
- B. 1
- C. 2
- D. 3
Answer: 2
Explanation: (xn – an) is divisible by (x – a) for all values of n ∴ (96 – 16) is divisible by (9 – 1) ⇒ (96 – 1) is divisible by 8 ⇒ On dividing (96 + 1) by 8, we get 2 as remainder.
400. (461 + 462 + 463 + 464) is divisible by
- A. 3
- B. 11
- C. 13
- D. 17
Answer: 17
Explanation: (461 + 462 + 463 + 464) = 461 (1 + 4 + 42 + 43) = 461 × 85, which is divisible by 17.
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