3. Number System 1 FOR CAT AND OMET EXAM

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1. Number System - 1 Method . Speed . Excellence

2. How many keystrokes are needed to type numbers from 1 to 1000? 3001 2893 2704 2890 None of these

3. While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers is

4. If A381 is divisible by 11, find the value of the smallest natural number A. (a) 5 (b) 6 (c) 7 (d) 9 (e) 8 If 381A is divisible by 9, find the value of smallest natural number A. (a) 5 (b) 5 (c) 7 (d) 6 (e) 11

5. Find the number of numbers between 200 and 300, both included, which are not divisible by 2, 3, 4 and 5. (a) 27 (b) 26 (c) 25 (d) 28 (e) 29 What is the value of M*N if M39048458N is divisible by 8 and 11, where M and N are single digit integers? (a) 6 (b) 16 (c) 24 (d) 0 (e) Cannot be determined How many numbers between 200 and 400 are divisible by 13? (a) 14 (b) 15 (c) 16 (d) 17 (e) None of these

6. 4 11 + 4 12 + 4 13 + 4 14 + 4 15 is divisible by which of the following? (a) 11 (b) 31 (c) 341 (d) All of these 3 25 + 3 26 + 3 27 + 3 28 is divisible by 11 (c) 16 25 (d) 30

7. How many of the integers 1, 2, … , 120, are divisible by none of 2, 5 and 7? CAT 2020 41 42 40 43

8. The number of integers x such that 0.25 < 2 x  < 200, and 2 x  + 2 is perfectly divisible by either 3 or 4, is [TITA] CAT 2018

9. Find the number of factors, sum of factors and the product of factors of 1800.

10. Find the number of odd and even factors of 1200.

11. Find the number of factors which are perfect squares of 10800.

12. If the given number is 10500 find the sum of the reciprocals of its factors.

13. The given number is 532400, find the number of factors of this number that are perfect cubes.

14. If there are 65 factors of a number N from 1 to √N, then find its total number of factors.

15. If both 11 2  and 3 3  are factors of the number a * 4 3  * 6 2  * 13 11 , then what is the smallest possible value of 'a’? 121 3267 363 33 None of the above

16. The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is CAT 2023

17. How many factors of 2 4  × 3 5  × 10 4  are perfect squares which are greater than 1? [TITA] CAT 2019

18. If (20!-19!)+k(19!-18!) = k(19!), what is the value of k? 17 2 17 × 19 19 2 17 × 18

19. Let n! = 1 × 2 × 3 × 4 × … × n for integer n ≥ 1. What could be the value of 1! + (2 × 2!) + (3 × 3!) + … + (10 × 10!) (11 × 11!) 11! 11! – 1 11! + 1

20. For some natural number n, assume that (15,000)! is divisible by (n!)!. The largest possible value of n is CAT 2022 5 7 4 6

21. Find the index of the greatest power (IGP) of 24 contained in 360!​

22. Find the index of the greatest power (IGP) of 10 which divides 1 25 × 2 24 × 3 23 × ... × 24 2 × 25 1 ? 56 55 34 45

23. What is the largest power of 2 in 32! +33! +34! +35! +...+90!?

24. The highest power of 7 that completely divides 80! is: (a) 12 (b) 13 (c) 14 (d) 15 (e) 11

25. How many trailing zeroes (zeroes at the end of the number) does 60! have? 14 12 10 8

26. Find the number of trailing zeroes in (2 5 × 5 2 )! 2 192 199 7

27. Express the following in the form of a + ib. (5i)(-3i/5) i 9 + i 19  

28. If z = x – iy and z 1/3 = p + iq , then ((x/p) + (y/q)) / (p 2 + q 2 ) is equal to? (a) -2 (b) -1 (c) 2 (d) 1

29. Thank You