NUMBER SYSTEM (CDS)

Total Questions: 101

71. What is the remainder when 4⁹⁶ is divided by 6 ? [2015 (I) Morning Shift ]

Correct Answer: (a) 4
Solution:4/6 then remainder = 4

72. The digit in the unit place of the product 81 × 82 × 83 × 84 ×...× 99 is [2015 (I) Morning Shift ]

Correct Answer: (a) 0
Solution:Product of unit digits = 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 0 × .... = 0
∴ Required digit in the unit place is 0.

73. What is the remainder obtained when 1421 × 1423 × 1425 is divided by 12 ? [2015 (I) Morning Shift ]

Correct Answer: (c) 3
Solution:

74. Consider the following statements [2015 (I) Morning Shift ]

I. The equation 1990x-173 y = 11 has no solution in integers for x and y.
II. The equation 3x - 12 y = 7 has no solution in integers for x and y.

Which of the above statement(s) is/are correct?

Correct Answer: (c) Both I and II
Solution:


75. What is the number of possible pairs of (P, Q), if the number 357P 25Q is divisible by both 3 and 5 ? [2015 (I) Morning Shift ]

Correct Answer: (a) 7
Solution:We know that, if the sum of the digits of a number is divisible by 3, then the number is divisible by 3 and if the digit at unit place is 5 or 0, then the number is divisible by 5.

76. The difference between the squares of two consecutive odd integers is always divisible by [2015 (I) Morning Shift ]

Correct Answer: (c) 8
Solution:Let the two consecutive odd integers be (2x - 1) and (2x + 1) respectively.
According to the question,
(2x + 1)² - (2x - 1)²
= (4x² + 1 + 4x) - (4x² + 1 - 4x)
= 4x² + 1 + 4x - 4x² - 1 + 4x
= 8x
Hence, the difference between the squares of two consecutive odd integers is always divisible by 8.

77. What is the maximum value of m, if the number N = 35 × 45 × 55 × 60 × 124 × 75 is divisible by 5ᵐ? [2015 (I) Morning Shift ]

Correct Answer: (c) 6
Solution:Here, 35 is divisible by 5 one time.
45 is divisible by 5 one time.
55 is divisible by 5 one time.
60 is divisible by 5 one time.
and 75 is divisible by 5 two times.
Hence, the maximum value of m is 6.

78. A light was seen regularly at an interval of 13 s. It was seen for the first time at 1 h 54 min 50 s (am) and the last time at 3 h 17 min 49 s (am). How many times was the light seen? [2015 (I) Morning Shift ]

Correct Answer: (c) 383
Solution:Number of light seen

79. Consider the following statements for the sequence of numbers; 11, 111, 1111, 11111,.... . [2015 (I) Morning Shift ]

I. Each number can be expressed in the form (4m + 3) where m is a natural number.
II. Some numbers are squares.

Which of the above statement(s) is/are correct?

Correct Answer: (a) Only I
Solution:Given numbers, 11, 111, 1111, 11111, ... ,

80. Consider the following statements [2015 (I) Morning Shift ]

I. There exists only one prime number p such that (17p + 1) is a square
II. If x is the product of 10 consecutive prime numbers starting from 2, then (x+1) is also a prime number.

Which of the above statement(s) is/are correct?

Correct Answer: (a) Only I
Solution: