Number Systems for Competitive Exam Success

Number Systems

Base Systems Digital Sum Divisibility Rules Factors & Multiples HCF & LCM Modular Arithmetic Remainders

Q. In a base-4 number system, what is the decimal equivalent of the number '132'?

A. 10
B. 12
C. 14
D. 16

Solution

'132' in base 4 is calculated as 1*4^2 + 3*4^1 + 2*4^0 = 16 + 12 + 2 = 30.

Correct Answer: B — 12

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Q. In a base-4 system, what is the sum of the numbers '12' and '21'?

A. 33
B. 33 in base-4
C. 33 in base-10
D. 30 in base-4

Solution

The sum of '12' and '21' in base-4 is '30' in base-4.

Correct Answer: D — 30 in base-4

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Q. In a base-4 system, what is the sum of the numbers 3 and 2?

A. 11
B. 12
C. 13
D. 20

Solution

In base-4, the sum of 3 and 2 is 11 (3 + 2 = 5 in decimal, which is 11 in base-4).

Correct Answer: A — 11

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Q. In a base-5 number system, what is the decimal equivalent of the base-5 number 243?

A. 35
B. 60
C. 55
D. 50

Solution

To convert from base-5 to decimal, multiply each digit by 5 raised to the power of its position (from right to left, starting at 0). Thus, 2*5^2 + 4*5^1 + 3*5^0 = 2*25 + 4*5 + 3*1 = 50 + 20 + 3 = 73.

Correct Answer: A — 35

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Q. In a base-5 number system, what is the decimal equivalent of the number '243'?

A. 60
B. 61
C. 62
D. 63

Solution

2*5^2 + 4*5^1 + 3*5^0 = 50 + 20 + 3 = 73.

Correct Answer: C — 62

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Q. In a base-5 number system, what is the sum of '23' and '14'?

A. 42
B. 32
C. 41
D. 30

Solution

In base-5, '23' is 2*5 + 3 = 13 and '14' is 1*5 + 4 = 9. Their sum is 22 in decimal, which is '42' in base-5.

Correct Answer: A — 42

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Q. In a base-5 number system, what is the sum of '34' and '21'?

A. 100
B. 110
C. 120
D. 130

Solution

In base-5, '34' is 3*5 + 4 = 19 and '21' is 2*5 + 1 = 11. Their sum is 30, which is '100' in base-5.

Correct Answer: A — 100

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Q. In a base-5 numeral system, what is the decimal equivalent of the base-5 number 243? (2023)

A. 60
B. 50
C. 40
D. 30

Solution

The base-5 number 243 is calculated as 2*5^2 + 4*5^1 + 3*5^0 = 50 + 20 + 3 = 60 in decimal.

Correct Answer: A — 60

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Q. In a base-5 numeral system, what is the decimal equivalent of the number 243?

A. 60
B. 61
C. 62
D. 63

Solution

The base-5 number 243 converts to decimal as 2*5^2 + 4*5^1 + 3*5^0 = 50 + 20 + 3 = 73.

Correct Answer: B — 61

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Q. In a base-5 numeral system, what is the representation of the decimal number 10?

A. 20
B. 22
C. 10
D. 15

Solution

In base-5, the decimal number 10 is represented as 20.

Correct Answer: A — 20

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Q. In a base-5 numeral system, what is the representation of the decimal number 12?

A. 22
B. 23
C. 24
D. 21

Solution

In base-5, the decimal number 12 is represented as 22 (2*5^1 + 2*5^0 = 12).

Correct Answer: B — 23

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Q. In a base-5 numeral system, what is the representation of the decimal number 7?

A. 12
B. 22
C. 14
D. 20

Solution

In base-5, the decimal number 7 is represented as 12, since 1*5^1 + 2*5^0 = 5 + 2 = 7.

Correct Answer: A — 12

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Q. In a base-5 numeral system, which of the following digits is not valid?

A. 0
B. 1
C. 2
D. 5

Solution

In a base-5 numeral system, valid digits are 0, 1, 2, 3, and 4; thus, 5 is not valid.

Correct Answer: D — 5

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Q. In a base-6 number system, what is the product of '12' and '3'?

A. 36
B. 40
C. 42
D. 44

Solution

'12' in base-6 is 1*6 + 2 = 8. '3' is 3. Their product is 8*3 = 24, which is '42' in base-6.

Correct Answer: C — 42

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Q. In a base-6 number system, what is the product of '24' and '13'?

A. 100
B. 110
C. 120
D. 130

Solution

'24' in base-6 is 2*6 + 4 = 16 and '13' is 1*6 + 3 = 9. Their product is 16*9 = 144, which is '100' in base-6.

Correct Answer: A — 100

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Q. In a base-6 number system, what is the product of '5' and '2'?

A. 10
B. 12
C. 14
D. 16

Solution

In base-6, '5' * '2' = 10 in decimal, which is '12' in base-6.

Correct Answer: B — 12

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Q. In a base-7 system, what is the base-10 equivalent of the base-7 number 345?

A. 150
B. 170
C. 180
D. 190

Solution

To convert from base-7 to decimal, calculate 3*7^2 + 4*7^1 + 5*7^0 = 3*49 + 4*7 + 5 = 147 + 28 + 5 = 180.

Correct Answer: B — 170

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Q. In a base-8 number system, what is the product of '7' and '5'?

A. 35
B. 37
C. 40
D. 42

Solution

7*5 = 35 in decimal, which is '43' in base-8.

Correct Answer: C — 40

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Q. In a binary system, what is the result of the operation '110 + 101'?

A. 1011
B. 1111
C. 1001
D. 1100

Solution

'110' (6 in decimal) + '101' (5 in decimal) = '1011' (11 in decimal).

Correct Answer: A — 1011

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Q. In a certain base, the number 'A3' represents 163 in decimal. What is the base of this number system?

A. 10
B. 11
C. 12
D. 13

Solution

Let the base be 'b'. Then, A3 = 10*b + 3 = 163. Solving gives b = 12.

Correct Answer: C — 12

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Q. In a certain division problem, if the dividend is 123 and the divisor is 10, what is the remainder?

A. 3
B. 2
C. 1
D. 0

Solution

The remainder when 123 is divided by 10 is 3, as 123 = 10*12 + 3.

Correct Answer: A — 3

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Q. In a certain division problem, if the dividend is 37 and the divisor is 5, what is the remainder? (2023)

A. 2
B. 3
C. 4
D. 1

Solution

When 37 is divided by 5, it goes 7 times (7*5=35) with a remainder of 2 (37-35=2).

Correct Answer: B — 3

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Q. In a certain modular arithmetic system, if 7 is congruent to 3 modulo 4, which of the following statements is true?

A. 7 - 3 is divisible by 4
B. 7 + 3 is divisible by 4
C. 7 * 3 is divisible by 4
D. 7 / 3 is divisible by 4

Solution

In modular arithmetic, if a ≡ b (mod m), then (a - b) is divisible by m. Here, 7 - 3 = 4, which is divisible by 4.

Correct Answer: A — 7 - 3 is divisible by 4

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Q. In a certain modular arithmetic system, if 7 is congruent to 3 modulo n, what can be inferred about n?

A. n must be greater than 4
B. n must be a prime number
C. n must be less than 4
D. n must be equal to 4

Solution

Since 7 - 3 = 4, n must be a divisor of 4. Therefore, n can be 1, 2, or 4, but for the congruence to hold, n must be greater than 4.

Correct Answer: A — n must be greater than 4

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Q. In a certain number system, the base is 5. What is the decimal equivalent of the number 243 in base 5?

A. 35
B. 60
C. 62
D. 50

Solution

To convert from base 5 to decimal, calculate: 2*5^2 + 4*5^1 + 3*5^0 = 2*25 + 4*5 + 3*1 = 50 + 20 + 3 = 73.

Correct Answer: C — 62

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Q. In a certain number system, the number '101' represents the decimal number 5. What does '111' represent in the same system?

A. 6
B. 7
C. 8
D. 9

Solution

'101' in binary is 5, so '111' in binary is 7.

Correct Answer: B — 7

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Q. In a certain number system, the number '101' represents the decimal number 5. What does '110' represent in the same system?

A. 6
B. 7
C. 8
D. 9

Solution

'101' in binary is 5 in decimal. '110' in binary is 6 in decimal.

Correct Answer: B — 7

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Q. In a certain number system, the number '123' represents the decimal number 27. What is the base of this number system?

A. 3
B. 4
C. 5
D. 6

Solution

In base 5, '123' = 1*25 + 2*5 + 3 = 27.

Correct Answer: C — 5

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Q. In a certain number system, the number '210' represents the decimal number 42. What is the decimal equivalent of '102' in the same system?

A. 20
B. 21
C. 22
D. 23

Solution

'210' in base 3 is 2*3^2 + 1*3^1 + 0*3^0 = 18 + 3 + 0 = 21. Therefore, '102' in base 3 is 1*3^2 + 0*3^1 + 2*3^0 = 9 + 0 + 2 = 11.

Correct Answer: A — 20

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Q. In a certain number system, the number 'A' represents 10. What is the value of 'B' if 'AB' represents 11?

A. 1
B. 2
C. 3
D. 4

Solution

'AB' means A*base + B = 11. Since A=10, we have 10*base + B = 11, so B=1.

Correct Answer: A — 1

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Showing 271 to 300 of 618 (21 Pages)

Number Systems MCQ & Objective Questions

Understanding number systems is crucial for students preparing for various exams in India. Mastering this topic not only enhances your mathematical skills but also boosts your confidence in tackling objective questions. Practicing MCQs related to number systems helps in identifying important questions and solidifying your exam preparation strategy.

What You Will Practise Here

Types of number systems: Natural, Whole, Integers, Rational, and Irrational numbers
Conversion between different number systems: Decimal, Binary, Octal, and Hexadecimal
Arithmetic operations in various number systems
Properties of numbers: Even, Odd, Prime, and Composite numbers
Understanding place value and significance in different bases
Common number system problems and their solutions
Real-world applications of number systems in technology and computing

Exam Relevance

Number systems are a fundamental part of the curriculum for CBSE, State Boards, NEET, and JEE. Questions related to this topic frequently appear in both objective and subjective formats. Students can expect to encounter problems that require conversions between bases, operations on numbers in different systems, and theoretical questions about properties of numbers. Familiarity with common question patterns will significantly enhance your performance in these exams.

Common Mistakes Students Make

Confusing the conversion process between different number systems
Overlooking the significance of place value in non-decimal systems
Misapplying arithmetic operations when dealing with binary or hexadecimal numbers
Ignoring the properties of numbers, leading to incorrect answers in problem-solving

FAQs

Question: What are the different types of number systems I should know for exams?
Answer: You should be familiar with natural numbers, whole numbers, integers, rational numbers, and irrational numbers, as these are commonly tested.

Question: How can I effectively practice number systems for my exams?
Answer: Regularly solving Number Systems MCQ questions and objective questions with answers will help reinforce your understanding and improve your speed.

Start solving practice MCQs today to test your understanding of number systems and boost your exam readiness. Remember, consistent practice is the key to success!

Number Systems

Number Systems MCQ & Objective Questions

What You Will Practise Here

Exam Relevance

Common Mistakes Students Make

FAQs

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