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Number System Test - 5

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Number System Test - 5
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  • Question 1
    3 / -1

    If the difference of (1025−7) and (1024+x) is divisible by 3 then x is equal to :

    Solution

    Given (1025−7)−(1024+x) is divisible by 3
    1025−7−1024−x
    1024(10−1)−7−x
    9.1924−7−x
    If divisible by 3 value of x=2

    Therefore, the correct option is B

  • Question 2
    3 / -1

    Convert  in p/q form

    Solution

    ► All the digits written once = 1762125
    ► All the digits without Bar written once = 1762
    ► No. of digits with bar after decimal = 3
    ► No. of digits without bar after decimal = 1
    Rational form = 

  • Question 3
    3 / -1

    (a2 + 2ab + b2) will fall under which of the categories, a and b are Integers

    Solution

    ► There are 4 case in the same
    ► Case 1 : a and b both are odd

    ⇒ a2 + 2ab + b2
    ⇒ Odd2 + 2 x Odd x Odd +Odd2
    ⇒ Odd + Even + Odd
    ⇒ Even

    ►Case 2 : a and b both are even

    ⇒ a2 + 2ab + b2
    ⇒ Even2 + 2 × Even × Even + Even2
    ⇒ Even + Even + Even
    ⇒ Even

    ► Case 3 : a is odd and b is even

    ⇒ a2 + 2ab + b2
    ⇒ Odd2 + 2 x Odd x Even + Even2
    ⇒ Odd + Even + Even
    ⇒ Odd

    ► Case 4 : a is even and b is odd

    ⇒ a2 + 2ab + b2
    ⇒ Even2 + 2 × Even × Odd + Odd2
    ⇒ Even + Even + Odd
    ⇒ Odd

    ► In 2 Cases its Odd and in other 2 it is Even, so we cannot say anything about the answer. 

  • Question 4
    3 / -1

    If (6a + 12) is odd then 'a' would be, a is an Integer

    Solution

    ► There are two possibilities for a
    ► Case 1: a is even

    ⇒ 6a  = Even x Even = Even
    ⇒ 6a + 12 = Even + Even = Even

    ► Case 2: a is odd

    ⇒ 6a = Even x Odd = Even
    ⇒ 6a + 12 = Even + Even = Even

    ► In both the cases the answer is even, so the information provided is inconsistent.

  • Question 5
    3 / -1

    The last two digits in the multiplication of 122×123×125×127×129 is

    Solution

    Divide the given expression by 100 and the expression gets reduced to 22×23×25×27×29 . 50 would be the last two digits here.

    SO, the correct option is B

  • Question 6
    3 / -1

     

    (abc) is odd what would (a2 + b2 + c2) be, a, b and c are Integers.

     

    Solution

     

    ► For abc to be Odd the only case possible is none of them being Even i.e. a, b and c all three are Odd

    ⇒ a2 + b2 + c2
    ⇒ Odd2 + Odd2 + Odd2
    ⇒ Odd + Odd + Odd
    ⇒ Odd

    Vice versa can also be true

     

  • Question 7
    3 / -1

     

    How many prime numbers are there between 25 and 75?

     

    Solution

     

    ► There are 12 prime numbers between 25 and 75 as listed below: 

     

  • Question 8
    3 / -1

    Which is the largest Even Prime number?

    Solution

    ► There is only one Even prime number i.e. 2. 

  • Question 9
    3 / -1

    Which of the following is a prime number?

    Solution

    ► We know the property that a prime number when divided by 6 gives a remainder of 1 or 5.
    ► Option A : 8831 divided by 6 gives a remainder of 5
    ► Option B : 8829 divided by 6 gives a remainder of 3
    ► Option C : 8833 divided by 6 gives a remainder of 1, but is divisible by 11
    ► Option D : 8835 divided by 6 gives a remainder of 3

  • Question 10
    3 / -1

    If a and b are odd numbers, which of the following is surely even?

    Solution

    ► Option A : a2 + b2 = Odd2 + Odd2 = Odd + Odd = Even
    ► Option B  : a2 + 2b2 = Odd2 + Even × Odd2 = Odd + Even = Odd
    ► Option C ∶ 2a2 + b2 = Even × Odd2 + Odd2 = Even + odd = Odd
    ► Option D ∶ a2 + b2 + ab = Odd2 + Odd2 + Odd × Odd = Odd + Odd + Odd = Odd

  • Question 11
    3 / -1

    Sum of any 50 consecutive 4 digit Whole numbers would always be:

    Solution

    ► Sum of Any Consecutive 50 Whole Numbers

    ⇒ Sum of 25 Even Numbers + Sum of 25 Odd Numbers
    ⇒ Even + Odd
    ⇒ Odd

  • Question 12
    3 / -1

    The sum of first 10 Prime numbers is

    Solution

    ► First 10 Prime numbers are listed Below :

    ► Their sum = 129

  • Question 13
    3 / -1

    How many Prime numbers less than 1000 are divisible by 17?

    Solution

    ► Being Prime numbers any number would be divisible by 1 or itself.
    ► Since 17 is a prime number. It will be the only prime number less than 1000 which will be divisible by 17.

  • Question 14
    3 / -1

    How many Prime numbers less than 1000 are divisible by 16?

    Solution

    ► Being Prime numbers any number would be divisible by 1 or itself.
    ► Since 16 is not a prime number. There would be no other prime number divisible by 16. 

  • Question 15
    3 / -1

    How many pairs of consecutive odd prime numbers are there less than 200?

    Solution

    ► The List of such pairs is listed below:
    (3,5); (5,7); (11,13); (17,19); (29,31); (41,43); (59,61); (71,73), (101,103); (107,109); (137,139);  (149,151); (179,181); (191,193); (197,199) 

    ► ​15 Such pairs exist.

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