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Q1. If $(13)_b = (27)_4$, find the base b.
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Q2. A number is divisible by 87 if it is divisible by both 3 and 29. Find the largest 6-digit number divisible by 87.
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Q3. If a number is represented as 101 in base 5, what is its value in base 10?
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Q4. The number of times the prime factor 7 appears in the prime factorization of 1000! is:
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Q5. Which of the following is a proper divisor of 10?
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Q6. A number when divided by 9 leaves a remainder of 7. What is the remainder when the same number is divided by 3?
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Q7. What is the GCD of 2023 and 2024?
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Q8. The area of a parallelogram is 60 sq cm. If its base is 10 cm, what is its height?
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Q9. A bag contains 10 balls numbered from 1 to 10. What is the probability of picking a ball with a number greater than 8?
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Q10. Calculate 34 base 7 multiplied by 5 base 7.
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Q11. A watch which normally gains 2 minutes every hour was fast by 5 minutes at 8 AM on Sunday. When will it show the correct time again?
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Q12. If 121 = 1+2+1 = 4, 345 = 3+4+5 = 12, and 678 = 6+7+8 = 21, what is 901?
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Q13. What is the greatest common divisor of 12! and 13!?
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Q14. If $(100)_b \times (20)_b = (2000)_b$, which of the following is true about $b$?
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Q15. If $123_5 \times 24_5 = X_5$, then find the value of X.
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Q16. In a dataset of 100 observations, the mean is 50. If one observation of 70 is removed, and the sum of the remaining 99 observations is 4930, what is the new mean?
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Q17. If a number is multiplied by 5 and then 5 is added to it, the result is 30. What is the number?
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Q18. From a deck of 52 cards, 4 cards are drawn. What is the probability that exactly 2 are Aces?
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Q19. A, B, and C can do a piece of work in 10, 15, and 12 days respectively. They all started the work together, but A left after 3 days. In how many days will B and C complete the remaining work?
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Q20. Let $a$ and $b$ be positive integers such that $a < b$. If $b-a = 5$ and $ rac{1}{a} - rac{1}{b} < rac{1}{24}$. What is the minimum possible value of $a$?
