InterviewSolution
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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
By which of the following numbers should `3^(5)` be divided to obtain a remainder 3 ?A. 7B. 11C. 5D. 3 |
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Answer» Correct Answer - C `3^(5) -= 3` (mod x) `rArr 243 = 3` (mod x) `rArr 243 - 3` is divisible by x `rArr 240` is divisible by x. `:.` From the options, x = 5. |
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| 2. |
If `a+2 -= 3` (mod 6), then a is __________. |
| Answer» Correct Answer - 1 | |
| 3. |
If `63 -= 2` (mod a) and `a gt 1`, then a is __________. |
| Answer» Correct Answer - 61 | |
| 4. |
If `x^(3) -= x` (mod 3), then x can be __________.A. 2B. 5C. 4D. All of these |
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Answer» Correct Answer - D Given `x^(3) -= x` (mod 3) `x^(3) - x` is divisible by 3. `x(x^(2) - 1)` is divisible by 3. `(x-1) x(x+1)` is divisible by 3. `x-1, x, x +1` are three consecutive numbers. We know that the product of three consecutive numbers is always divisible by 3. `:.` For any integer of x (where `x ge 2`), the given relation is true. |
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| 5. |
`15 -= -3` (mod 9). (True/False) |
| Answer» Correct Answer - 1 | |
| 6. |
Now the time is `1 :30` pm. If I woke up 8 hours ago, then I woke up at_________.A. `4:30` pmB. `5:30` pmC. `3:30` pmD. `6:30` pm |
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Answer» Correct Answer - B Check from the options. |
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| 7. |
In order to enter her name in the Guinness Book of world records, Sangeeta started singing on Monday at 10.30 am. If the sings continuously for 36 hours then she will finish her singing onA. Tuesday at 10.30 amB. Wednesday at 10.30 amC. Tuesday at 10.30 pmD. Wednesday at 10.30 pm |
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Answer» Correct Answer - C Monday, 10.30 am to Tuesday, 10.30 am is 24 hours. Tuesday, 10.30 am to Tuesday, 10.30 pm = 12 10.30 hours. `:.` She will finish her singing on Tuesday night at 10.30 pm. |
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| 8. |
If `13 -= 3` (mod p), then p can beA. 2B. 5C. 10D. All of these |
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Answer» Correct Answer - D Given `13-=3` (mod p) `rArr 13 - 3` is divisible by p `rArr 10` is divisible by p i.e., p is a factor of 10. `:.` p = 1, 2 5 or 10 `:.` All the options are true. |
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| 9. |
If `x -= y` (mod m), then `6x - 5 -= 6y - 5` (mod m). (True/False). |
| Answer» Correct Answer - 1 | |
| 10. |
Which of the following is/are correct ?A. `5 o+_(2) 4 -= 17 ox_(5) 3` (mod 7)B. `6 o+_(4) 7 -= 19 ox_(9) 3` (mod 3)C. `9 o+_(7) 3 -= 8 ox_(7) 9` (mod 9)D. `5 o+_(2) 4 -= 17 ox_(5) 3` (mod 5) |
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Answer» Correct Answer - A `5 o+_(2) 4 -= 17 ox_(5) 3` (mod 7) 1 = 1 (mod 7) `:.` 1-1 is divisible by 7 (true). |
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| 11. |
Kishore reached his school on Monday at 8:30 am, and them immediately started on a tour to GOA. After `106(1)/(2)` hours, he reached his house. Then, Kishore reached his home onA. Saturday at 7 pm.B. Friday at 6pm.C. Saturday at 6 pm.D. Friday at 7 pm. |
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Answer» Correct Answer - D `106 (1)/(2)` hours = 4 days `+ 10(1)/(2)` hours. |
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| 12. |
In a certain month, the first Sunday falls on the fifth day of the month. In the same month, the fourth Sunday falls on the ________day. |
| Answer» Correct Answer - 26th | |
| 13. |
If the 1st January of a certain year, which was not a leap year, was a Thursday, then what day of the week was the 31st December of that year ?A. MondayB. ThursdayC. SundayD. Saturday |
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Answer» Correct Answer - B The given year is a non-leap year and contains 365 days. (The number of days from 1st January to 31st December is 365) `365 -= 1` (mod 7) The 365th day is equivalent to the first day of the year. `:.` But the first day of the week is Thursday. `:.` Hence, 31st December is Thursday. |
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| 14. |
The 2-1-2009 is a Friday. The fourth Sunday of January 2010 falls on the _________.A. 23rd dayB. 24th dayC. 25th dayD. 26th day |
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Answer» Correct Answer - B The number of days from 2-1-2009 to 1-1-2010 is 365 days `rArr 365 -= 1` (mod 7) `:.` 1-1-2010 is Friday (`:.` 2-1-2009 is Friday) `:.` 1st Sunday falls on 3-1-2010 `:.` 2nd Sunday falls on 10-1-2010 `:.` 3rd Sunday falls on 17-1-2010 `:.` 4th Sunday falls on 24-1-2010. |
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| 15. |
If `x + 10 -= 1` (mod 8), then x can be ________.A. 1B. 0C. 6D. 7 |
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Answer» Correct Answer - D Given x + 10 = 1 (mod 8) `rArr x + 10 - 1` is divisible by 8 `rArr x + 9` is divisible by 8 From the options, if x = 7 then 7 + 19, i.e., 16 is divisible by 8. |
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| 16. |
Anand started a work on Sunday at 9:30 am. He finished the work after 87 hours. Then the finished the work on______.A. Wednesday at 11 : 30 pmB. Thursday at 0 : 30 pmC. Wednesday at 0 : 30 amD. Thursday at 11 : 30 pm |
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Answer» Correct Answer - B 87 hours = 3 days + 15 hours. |
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| 17. |
Which of the following is/are correct ?A. `5 o+_(4) 3 -= 7 ox_(9) 8` (mod 5)B. `10 o+_(5) 4 -= 9 ox_(11) 9` (mod 11)C. `14 o+_(8) 8 -= 15 ox_(16) 12` (mod 4)D. `10 o+_(5) 4 -= 9 ox_(11) 9` (mod 10) |
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Answer» Correct Answer - B (i) Use the definitions of addition modulo m and multiplication modulo m. (ii) Substitute the values in the options in the given inequations. (iii) The point which satisfies the given inequations is the required point. |
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| 18. |
Fin dthe remainder when `2^(24)` is divided by 35.A. 2B. 31C. 1D. 29 |
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Answer» Correct Answer - C Use, if p is prime then `a^(p) -= a (mod p)`. |
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| 19. |
Which of the following is a common solution of `3x-= 2` (mod 5) and `4x -= 0` (mod 6) ?A. 9B. 4C. 6D. 3 |
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Answer» Correct Answer - A Given `3x -= 2` (mod 5) `rArr 3x - 2` is divisible by 5. `x - 4` or 9 or 14, or ...........satisfies the above relations. `4x -= 0` (mod 6) `rArr 4x` is divisible by 6. x = 3 or 9 or 15......... Satisfies the above relation. `:.` The common solution of `3x -= 2` (mod 5) and `4x -= 0` (mod 6) is 9. |
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| 20. |
If 1-1-2010 is a Friday, then the fifth Sunday of January, 2011 will fall on_______.A. 26th dayB. 27th dayC. 29th dayD. 30th day |
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Answer» Correct Answer - D 1-1-2010 is a Friday. 1-1-2011 is a Saturday. First Sunday in 2011 is 2nd January. |
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| 21. |
Which of the following are the common solutions of `3x -= 0` (mod 6) and `2x -= 0` (mod 4) ? (A) 0 (B) 2 (C) 4A. Both (A) and (B)B. Both (A) and (C)C. Both (B) and (C)D. All of (A), (B) and (C) |
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Answer» Correct Answer - D Verify whether the given options are common solutions are not. |
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| 22. |
Which of the following is a common solutions for `6x -= 0` (mod 8) and `8x -= 0` (mod 10) ? |
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Answer» Correct Answer - A Verify whether each option is a solutions of both the equations or not. |
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| 23. |
If `x -= y` (mod 2), then which of the following are correct ? (A) x is even and y is odd. (B) Both x and y are odd. (C) Both x and y are even.A. Only (C)B. Only (A)C. Both (B) and (C)D. Both (A) and (B) |
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Answer» Correct Answer - C If a-b is divisible by 2, then both a and b are either even number or odd numbers. |
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| 24. |
If `5x -= 2` (mod 3), then which of the following is a possible value of x?A. 3B. 315C. 0D. None of these |
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Answer» Correct Answer - D 15x is always divisible by 3. |
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| 25. |
Solve for x, if `5x -= 0` (mod 4). |
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Answer» Correct Answer - A Check from the options. |
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| 26. |
Find the remainder when `11^(12)` is divided by 7. |
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Answer» Correct Answer - B Use the concept of congruence modulo. |
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| 27. |
If x belongs to the set of residues modulo 10, then the common solution of `5 + x -= 0` (mod 3) and `6 + x -= 0` (mod 5) is ________.A. 1B. 2C. 4D. 5 |
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Answer» Correct Answer - C `Z_(10) = {0,1,2,3,4,5,6,7,8,9}` and `x in Z_(10) 5+x -= 0` (mod 3) `rArr x = 1, 4` and 7 `6 + x -= 0` (mod 5) `rArr x = 4` and 9 `:.` Common solution is x = 4. |
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| 28. |
If x belongs to the set of residues modulo 4 and `2 + x -= 5` (mod 4), then x = __________. |
| Answer» Correct Answer - 3 | |
| 29. |
If x belongs to the set of residues modulo 6 and `3 + x -= 2` (mod 6), then x = ________.A. 1B. 3C. 4D. 5 |
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Answer» Correct Answer - D `Z_(6) = {0,1,2,3,4,5}` and `x in Z_(6)`. Given, `3 + x -= 2` (mod 6) `rArr 3 + x - 2` is divisible by 6 `rArr x + 1` is divisible by 6 x = 5 satisfies the above conditions. |
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| 30. |
In te set of integers modulo 6, `35 ox_(6) 5` = _________. |
| Answer» Correct Answer - 1 | |
| 31. |
In the set of integers modulo 5, `16 ox_(5) 7` = ___________. |
| Answer» Correct Answer - 3 | |
| 32. |
In the set of integers modulo 6, `28 ox_(8) 2` =________. |
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Answer» Correct Answer - A Recall the concept of molecular mupltiplication. |
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| 33. |
In the set of integers modulo, 17, `19 o+_(17) 15` = _________. |
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Answer» Correct Answer - A The remainder when 19 + 15, i.e., 34 is divided by 17 is 0. |
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| 34. |
Find the remainder when `6^(11) - 6` is divided by 11.A. 5B. 1C. 0D. None of these |
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Answer» Correct Answer - C We have `a^(p) -= a` (mod p) where p is a prime number. 11 is prime number, `6^(11) -= 6` (mod 11) That is, `6^(11) - 6` is divisible by 11. The remainder when `6^(11)-6` is divided by 11 is zero. |
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| 35. |
If `x - 4 -= 8` (mod 5), then x is __________. |
| Answer» Correct Answer - 2 | |
| 36. |
If `a ox_(m) b = 1`, then b is called the reciprocal of under modulo m. Find the reciprocal of 8 under modulo 17. |
| Answer» Correct Answer - 3 | |
| 37. |
The largest two-digit number that satisfies `5x -= 6` (mod 4) is ________.A. 96B. 97C. 98D. 99 |
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Answer» Correct Answer - C Check from the options. |
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| 38. |
If `46 -= 11` (mod a), and a is a prime number, then find the greatest possible value of a. |
| Answer» Correct Answer - 7 | |
| 39. |
If x belongs to the set of residues modulo 6 and `5 + x -= 3` (mod 6), then find x. |
| Answer» Correct Answer - x = 4 | |
| 40. |
The largest single-digit number that satisfies `14x -= 4` (mod 3) is_______.A. 5B. 7C. 8D. 9 |
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Answer» Correct Answer - C Check from the options. |
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| 41. |
If x belongs to the set of residues modulo 4 and `6x - 3 -= -1` (mod 4), then find x. |
| Answer» Correct Answer - 1 or 3 | |
| 42. |
Find the remainder when `5^(18)` is divided by 19.A. 1B. 4C. 11D. 17 |
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Answer» Correct Answer - A Start with the step `5^(2) -= 6` (mod 19) and proceed. |
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| 43. |
Find x, if `9x -= 2` (mod 7).A. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - A Given, `9x -= 2` (mod 7) `rArr 9x -2` is divisible by 7. From the options, if x = 1, then 9 - 2 is divisible by 7. |
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| 44. |
In the set of integers modulo 9, `15 ox_(9) 10` = __________.A. 3B. 6C. 0D. 1 |
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Answer» Correct Answer - B The remainder when `15 xx 10`, i.e., 150 is divided by 9 is 6. |
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| 45. |
Find the remainder when `3^(31)` is divided by 31. |
| Answer» Correct Answer - 2 | |
| 46. |
How many two digit numbers satisfy the equations `3x -= 5` (mod 7)? |
| Answer» Correct Answer - 4 | |
| 47. |
Find the remainder when `3^(19)` is divided by 19.A. 3B. 15C. 16D. 19 |
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Answer» Correct Answer - A We have `a^(p) -= a` (mod p), where p is a prime number. When `3^(19)` is divided by 19, the remainder is 3. |
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| 48. |
Find the remainder when `8^(15)` is divided by 5. |
| Answer» Correct Answer - 2 | |
| 49. |
Find the remainder when `(26)^(31)` is divided by 31. |
| Answer» Correct Answer - 31 | |
| 50. |
Find the remainder when `5^(97)` is divided by 97.A. 5B. 97C. 92D. 100 |
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Answer» Correct Answer - A We know that, when p is a prime number `a^(p) -= a` (mod p). |
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