Number System (Test 4)

Cat Entrance Exams : Mathematics Or Quantitative Aptitude

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Number System

Number System
| Number System |
Q.1
The difference between a positive proper fraction and its reciprocal is 9/20. The fraction is:
A. 3/5
B. 3/10
C. 4/5
D. 4/3
Answer : Option C
Explaination / Solution:

Let the required fraction be x. Then1x =9
x20

   1 - x2=9
x20

     20 - 20x2 = 9x

     20x2 + 9x - 20 = 0

     20x2 + 25x - 16x - 20 = 0

     5x(4x + 5) - 4(4x + 5) = 0

     (4x + 5)(5x - 4) = 0

x =4
5


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Q.2
On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8, what will be the remainder ?
A. 4
B. 5
C. 6
D. 7
Answer : Option B
Explaination / Solution:

Formula: (Divisor*Quotient) + Remainder = Dividend.

Soln:

(56*Q)+29 = D -------(1)

D%8 = R -------------(2)

From equation(2),

((56*Q)+29)%8 = R.

=> Assume Q = 1.

=> (56+29)%8 = R.

=> 85%8 = R

=> 5 = R.


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Q.3
If n is a natural number, then (6n2 + 6n) is always divisible by:
A. 6 only
B. 6 and 12 both
C. 12 only
D. by 18 only
Answer : Option B
Explaination / Solution:

(6n2 + 6n) = 6n(n + 1), which is always divisible by 6 and 12 both, since n(n + 1) is always even.

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Q.4
107 x 107 + 93 x 93 = ?
A. 19578
B. 19418
C. 20098
D. 21908
E. None of these
Answer : Option C
Explaination / Solution:

107 x 107 + 93 x 93= (107)2 + (93)2
= (100 + 7)2 + (100 - 7)2
= 2 x [(100)2 + 72]       [Ref: (a + b)2 + (a - b)2 = 2(a2+ b2)]
= 20098

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Q.5
What will be remainder when (6767 + 67) is divided by 68 ?
A. 1
B. 63
C. 66
D. 67
Answer : Option C
Explaination / Solution:

(xn + 1) will be divisible by (x + 1) only when n is odd.

(6767 + 1) will be divisible by (67 + 1)

(6767 + 1) + 66, when divided by 68 will give 66 as remainder.


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Q.6
On dividing a number by 5, we get 3 as remainder. What will the remainder when the square of the this number is divided by 5 ?
A. 0
B. 1
C. 2
D. 4
Answer : Option D
Explaination / Solution:

Let the number be x and on dividing x by 5, we get k as quotient and 3 as remainder.

    x = 5k + 3

    x2 = (5k + 3)2

   = (25k2 + 30k + 9)

   = 5(5k2 + 6k + 1) + 4

On dividing x2 by 5, we get 4 as remainder.


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Q.7
How many 3-digit numbers are completely divisible 6 ?
A. 149
B. 150
C. 151
D. 166
Answer : Option B
Explaination / Solution:

3-digit number divisible by 6 are: 102, 108, 114,... , 996

This is an A.P. in which a = 102, d = 6 and l = 996

Let the number of terms be n. Then tn = 996.

 a + (n - 1)d = 996

 102 + (n - 1) x 6 = 996

 6 x (n - 1) = 894

 (n - 1) = 149

 n = 150

 Number of terms = 150.


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Q.8
How many natural numbers are there between 23 and 100 which are exactly divisible by 6 ?
A. 8
B. 11
C. 12
D. 13
E. None of these
Answer : Option D
Explaination / Solution:

Required numbers are 24, 30, 36, 42, ..., 96

This is an A.P. in which a = 24, d = 6 and l = 96

Let the number of terms in it be n.

Then tn = 96    a + (n - 1)d = 96

 24 + (n - 1) x 6 = 96

 (n - 1) x 6 = 72

 (n - 1) = 12

 n = 13

Required number of numbers = 13.


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Q.9
How many of the following numbers are divisible by 3 but not by 9 ? 2133, 2343, 3474, 4131, 5286, 5340, 6336, 7347, 8115, 9276
A. 5
B. 6
C. 7
D. None of these
Answer : Option B
Explaination / Solution:

Marking (/) those which are are divisible by 3 by not by 9 and the others by (X), by taking the sum of digits, we get:s

2133  9 (X)

2343  12 (/)

3474  18 (X)

4131  9 (X)

5286  21 (/)

5340  12 (/)

6336  18 (X)

7347  21 (/)

8115  15 (/)

9276  24 (/)

Required number of numbers = 6.


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Q.10
(963 + 476)2 + (963 - 476)2= ?
(963 x 963 + 476 x 476)
A. 1449
B. 497
C. 2
D. 4
E. None of these
Answer : Option C
Explaination / Solution:

Given Exp. =(a + b)2 + (a - b)2=2(a2 + b2)= 2
(a2 + b2)(a2 + b2)

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    Mathematics or Quantitative Aptitude