NUMBER SYSTEM (CDS)

Total Questions: 101

41. (NPᵖ⁻¹-1) is a multiple of p, if N is prime to p and p is a [2017 (I) Morning Shift ]

Correct Answer: (a) prime number
Solution:According to Fermat's theorem –
If p is a prime number and n is prime to p,
then (nᵖ⁻¹ - 1) is divisible by p.

42. Consider the following statements [2017 (I) Morning Shift ]

1. Of two consecutive integers one is even.
2. Square of an odd integer is of the form 8n + 1

Which of the above statements is / are correct?

Correct Answer: (c) Both 1 and 2
Solution:1. Any integer number is either even or odd. Let first number be odd i.e. 2n + 1, then next consecutive number will be 2n + 2 = 2(n + 1) = 2m = even.

If we take first number as even, then next number will be 2n + 1 = odd.
So, one number is always even.

2. 1² = 1 = 8 × 0 + 1
3² = 9 = 8 × 1 + 1
5² = 25 = 8 × 3 + 1 and so on
∴ Square of any odd integer is always of the form 8n + 1
Hence, both statements are correct.

43. What would be the maximum value of Q in the equation 5P9 + 3R7 + 2Q8 = 1114 ? [2016 (II) Evening Shift ]

Correct Answer: (a) 9
Solution:Given,
P59 + 3R7 + 2Q8 = 1114
From above equation, we get
P + R + Q + 2 = 11
⇒ P + R + Q = 9
Hence, the maximum value of Q is 9.

44. A boy saves ₹4.65 daily. What is the least number of days in which he will be able to save an exact number of rupees? [2016 (II) Evening Shift ]

Correct Answer: (b) 20
Solution:From the given options, 20 is the least number of days in which he will be able to save an exact number of rupees because 4.65 × 20 = 93, which is an exact number of rupees.

45. What is the remainder when 2¹⁰⁰ is divided by 101 ? [2016 (II) Evening Shift ]

Correct Answer: (a) 1
Solution:Using Fermat's little theorem, aᵖ⁻¹ = 1 (mod p) for prime p.
Here, a = 2 and p = 101 given
⇒ 2¹⁰⁰ =1 mod (101)
∵ 101 is prime and does not divide 2.
So, the answer is 1.

46. Which of the following is correct in respect of the number 1729 ? [2016 (II) Evening Shift ]

Correct Answer: (c) It can be written as the sum of the cubes of two positive integers in two ways only
Solution:We have, 1729 = 1728 +1= (12)³ + (1)³ or 1729 = 1000 + 729 = (10)³ + (9)³
∴ 1729 can be written as sum of the cubes of two positive integers in two ways only.

47. Consider the following statements in respect of positive odd integers x and y. [2016 (II) Evening Shift ]

I. x² + y² is even integer.
II. x² + y² is divisible by 4.

Which of the above statements is/are correct?

Correct Answer: (a) I only
Solution:We have,

48. 2¹²² + 4⁶² + 8⁴² + 4⁶⁴ + 2¹³⁰ is divisible by which one of the following integers? [2016 (II) Evening Shift ]

Correct Answer: (d) 11
Solution:We have,

49. Let S be a set of first ten natural numbers. What is the possible number of pairs (a, b) where a, b ∈ S and a ≠ b such that the product ab (>12) leaves remainder 4 when divided by 12 ? [2016 (II) Evening Shift ]

Correct Answer: (c) 8
Solution:The numbers (> 12) which divided by 12 leaves remainder 4, are 16, 28, 40, 52, 64, 76, 88 and 100 and so on.
As (a, b) belongs to set of first ten natural numbers and a ≠ b.
Hence, possible product ab (> 12) leaves remainder 4 when divided by 12 are 16, 28 and 40.
Thus, set of such pairs (a, b) are (2, 8), (8, 2), (4, 7), (7, 4), (5, 8), (8, 5), (10, 4) and (4, 10)
Hence, the possible number of such pairs are 8.

50. What is the remainder, when 13⁵ + 14⁵ + 15⁵ + 16⁵ is divided by 29? [2016 (II) Evening Shift ]

Correct Answer: (d) 0
Solution:13⁵ + 16⁵ is divisible by 13 + 16 = 29 (as 5 is odd)