Level 1
Number Theory MCQs set 3 for OTS Revenue Department Posts Mathematics — 20 solved questions.
Q1. Two numbers 4242 and 2903 when divided by a certain number of three digits, leave the same remainder. Find the number?
Answer: 103
Explanation: The required number divides 4242−2903 = 1339. Factorising 1339 = 13 × 103; the three-digit factor is 103.
Q2. The least number of four digits which is divisible by 4, 6, 8 and 10 is_________?
Answer: 1080
Explanation: LCM of 4, 6, 8, 10 = 120; smallest four-digit multiple of 120 = 120 × 9 = 1080.
Q3. The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively, is_________?
Answer: 127
Explanation: Find HCF of (1657−6)=1651 and (2037−5)=2032. Applying the Euclidean algorithm gives HCF=127.
Q4. What is the smallest number, which when divided by 3,8 and 15 leaves the remainder 1,6,13 respectively?
Answer: 118
Explanation: The deficit (divisor−remainder) is constant at 2 for each divisor. LCM(3,8,15) = 120. Answer = 120−2 = 118.
Q5. In a division, problem the divisor is 4 times the quotient and 3 times the remainder, if remained is 4, then the dividend is?
Answer: 40
Explanation: Remainder = 4, divisor = 3×4 = 12, quotient = 12/4 = 3; dividend = 12×3 + 4 = 40.
Q6. How many two-digit numbers are there that are divisible by 11?
Answer: 9
Explanation: Two-digit multiples of 11 are: 11, 22, 33, 44, 55, 66, 77, 88, 99 - a total of 9 numbers.
Q7. Which one of the following is not a prime number?
Answer: 91
Explanation: 91 = 7 × 13, so it has factors other than 1 and itself, making it a composite number, not prime.
Q8. The H.C.F of two numbers is 23 and the other two factors of their L.C.M are 13 and 14. The larger of the two numbers is___________?
Answer: 322
Explanation: The two numbers are 23 × 13 = 299 and 23 × 14 = 322; the larger is 322.
Q9. Find the least number which when divide by 2, 3, 4, 5 and 6 leaves 1, 2, 3, 4 and 5 as remainders respectively, but when divided by 7 leaves no remainder?
Answer: 119
Explanation: LCM(2,3,4,5,6) = 60; numbers of the form 60k−1 that are divisible by 7: 60×2−1 = 119 = 7×17.
Q10. Find the lowest common multiple of 24, 36 and 40. LCM
Answer: 360
Explanation: 24=2³×3, 36=2²×3², 40=2³×5. LCM = 2³×3²×5 = 360. Show the calculation clearly when solving similar quantitative items.
Q11. The least number which, when increased by 3, is completely divisible by 8, 12 and 18 is_______?
Answer: 69
Explanation: LCM(8, 12, 18) = 72; the required number is 72 − 3 = 69, which when increased by 3 equals 72.
Q12. The least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3 is:
Answer: 23
Explanation: LCM(5, 6, 4, 3) = 60; 2497 ÷ 60 leaves remainder 37; adding 60 − 37 = 23 makes 2520, which is divisible by all four numbers.
Q13. What are Prime Numbers?
Answer: Which can be divided by Number 1 & by itself Number
Explanation: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Q14. Find the least multiple of 13 which when divided by 6, 8 and 12 leaves 5, 7 and 11 as remainders respectively?
Answer: 143
Explanation: LCM(6,8,12) = 24; required number = 24k-1 and divisible by 13. Testing: 24×6-1 = 143 = 11×13. So 143 is the answer.
Q15. Find the least square number which is divisible by 10, 12, 15 and 18?
Answer: 900
Explanation: LCM(10,12,15,18) = 180 = 2²×3²×5. To make it a perfect square, multiply by 5 to get 900 = 2²×3²×5².
Q16. Factorize 3x²y² - 14xy + 16:
Answer: (xy-2)(3xy-8)
Explanation: The correct value is (xy-2)(3xy-8). Apply the formula or arithmetic step shown in the question and
Q17. The product of two digits number is 2160 and their HCF is 12. The numbers are________?
Answer: (36,60)
Explanation: HCF=12, so numbers = 12a and 12b where a,b are co-prime. 144ab=2160 → ab=15. Co-prime pairs: (1,15) and (3,5). Only 2-digit pair is (36, 60).
Q18. Find the greatest number which, while dividing 19, 83 and 67, gives a remainder of 3 in each case?
Answer: 16
Explanation: Subtracting the remainder 3 from each: 16, 80, 64; GCF(16,80,64) = 16.
Q19. The least square number which is divisible by 8, 12 and 18 is________?
Answer: 144
Explanation: LCM(8,12,18) = 72 = 2³×3²; the smallest perfect square multiple is 144 = 2⁴×3² = 12².
Q20. Find the least number of five digits which is exactly divisible by 12, 15 and 18?
Answer: 10080
Explanation: LCM(12, 15, 18) = 180. The smallest 5-digit multiple of 180 is 180×56 = 10080.