Level 1
Number Theory MCQs set 2 for NTS Pakistan Railways Mathematics — 20 solved questions.
Q1. Factorize completely: 12x²y + 12xy². Algebra
Answer: 12xy(x+y)
Explanation: The correct value is 12xy(x+y). Apply the formula or arithmetic step shown in the question and
Q2. How many two-digit numbers are divisible by 9?
Answer: 10
Explanation: Two-digit multiples of 9 are 18, 27, 36, 45, 54, 63, 72, 81, 90, 99 - a total of 10 numbers.
Q3. Three consecutive odd integers are in increasing order such that the sum of the last two integers is 13 more than the first integer. Find the three integers?
Answer: 7, 9, 11
Explanation: For odd integers n, n+2, n+4: (n+2)+(n+4) = n+13 gives n = 7, so the integers are 7, 9, 11.
Q4. If (461 + 462 + 463 + 464) is divisible by ?
Answer: None of these
Explanation: The correct value is None of these. Apply the formula or arithmetic step shown in the question and
Q5. In the sequence a_n = 3n+2, what is the LCM of the 6th and 7th terms? (Note: the symbol _ indicates subscript) Arithmetic Progression
Answer: 460
Explanation: 6th term = 3(6)+2 = 20; 7th term = 3(7)+2 = 23; LCM(20,23) = 460.
Q6. The smallest number when increased by " 1 " is exactly divisible by 12, 18, 24, 32 and 40 is__________?
Answer: 1439
Explanation: LCM(12, 18, 24, 32, 40) = 1440; the required number is 1440 − 1 = 1439.
Q7. If n is a natural number, then 92n- 42n is always divisible by?
Answer: Both A '& B
Explanation: The correct value is Both A '& B. Apply the formula or arithmetic step shown in the question and
Q8. Out of first 20 natural numbers, one number is selected at random. The probability that it is either an even number or a prime number is_____________?
Answer: 17/20
Explanation: Even numbers from 1-20: 10; primes from 1-20: 8; union = 10+8−1 (for 2) = 17; probability = 17/20.
Q9. The least number which when diminished by 7 is divisible by 21, 28, 36 and 45 is________?
Answer: 1267
Explanation: LCM(21, 28, 36, 45) = 1260. The required number is 1260 + 7 = 1267.
Q10. Find the highest common factor of 36 and 84.
Answer: 12
Explanation: HCF(36, 84): 84 = 2×36+12; 36 = 3×12+0. So HCF = 12.
Q11. The smallest number which when divided by 20, 25, 35 and 40 leaves a remainder of 14, 19, 29 and 34 respectively is________?
Answer: 1394
Explanation: Each remainder is 6 less than the divisor, so the required number = LCM(20,25,35,40)−6 = 1400−6 = 1394.
Q12. HCF and LCM two numbers are 12 and 396 respectively. If one of the numbers is 36, then the other number is_______?
Answer: 132
Explanation: Using HCF × LCM = Product of two numbers: 12 × 396 = 36 × other number, giving other number = 4752 ÷ 36 = 132.
Q13. HCF of 3/16, 5/12, 7/8 is__________?
Answer: Jan-48
Explanation: HCF of fractions = HCF of numerators / LCM of denominators = HCF(3,5,7)/LCM(16,12,8) = 1/48.
Q14. Which of the following numbers is 423 divisible by?
Answer: 47
Explanation: Dividing 423 by 47 gives exactly 9 with no remainder, confirming 423 = 47 × 9.
Q15. The greatest number of four digits that have 144 for their HCF is___________?
Answer: 9936
Explanation: The largest multiple of 144 that is a four-digit number is 144 × 69 = 9936.
Q16. Product of two co-prime numbers is 117. Their L.C.M should be__________?
Answer: 117
Explanation: For co-prime numbers, HCF = 1. Since LCM × HCF = product of the two numbers, LCM = product/1 = 117.
Q17. Masood invests a part of Rs. 12000 in 12% stock at Rs. 120 and the remainder in 15% stock at Rs. 125. If his total dividend per annum is Rs. 1360, how much does he invest in 12% stock at Rs. 120?
Answer: Rs4,000
Explanation: Let x be invested in 12% stock at Rs.120; dividend = x × (12/120) = 0.1x. The rest earns (12000−x) × (15/125) = 0.12(12000−x). Setting total = 1360: 0.1x + 1440 − 0.12x = 1360, so x = Rs.4000.
Q18. Find the greatest number which is such that when 697, 909 and 1227 are divided by it, the remainders are all the same?
Answer: 106
Explanation: The required number is the GCD of the differences: 909−697=212, 1227−909=318. GCD(212, 318) = 106.
Q19. Find the greatest number that, while dividing 47, 215 and 365, gives the same remainder in each case?
Answer: 6
Explanation: The required number divides all differences: 215−47=168, 365−215=150. GCD(168, 150) = 6.
Q20. Find the lowest 4-digit number which when divided by 3, 4 or 5 leaves a remainder of 2 in each case?
Answer: 1022
Explanation: LCM(3,4,5) = 60. The lowest 4-digit multiple of 60 is 1020. Adding the remainder of 2 gives 1022.