Q. If a ≡ 2 (mod 5) and b ≡ 3 (mod 5), what is the value of (a * b) mod 5?
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Solution
a * b ≡ 2 * 3 ≡ 6 (mod 5), which is equivalent to 1.
Correct Answer:
B
— 1
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Q. If a ≡ 3 (mod 7) and b ≡ 5 (mod 7), what is a + b (mod 7)?
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Solution
a + b ≡ 3 + 5 ≡ 8 ≡ 1 (mod 7).
Correct Answer:
A
— 1
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Q. If a ≡ b (mod m) and c ≡ d (mod m), which of the following is necessarily true?
A.
a + c ≡ b + d (mod m)
B.
a * c ≡ b * d (mod m)
C.
a - c ≡ b - d (mod m)
D.
All of the above
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Solution
All operations (addition, multiplication, and subtraction) preserve congruence in modular arithmetic.
Correct Answer:
D
— All of the above
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Q. If a ≡ b (mod m) and c ≡ d (mod m), which of the following is true?
A.
a + c ≡ b + d (mod m)
B.
a - c ≡ b - d (mod m)
C.
a * c ≡ b * d (mod m)
D.
All of the above
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Solution
All operations (addition, subtraction, multiplication) maintain congruence under modular arithmetic.
Correct Answer:
D
— All of the above
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Q. If the base of a number system is increased by 1, how does the representation of the number '10' change?
A.
It remains the same
B.
It becomes 11
C.
It becomes 1
D.
It becomes 2
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Solution
In base 'b', '10' represents 'b'. In base 'b+1', it becomes '11'.
Correct Answer:
B
— It becomes 11
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Q. If the base-2 number 1101 is converted to base-10, what is the result?
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Solution
To convert from base-2 to decimal, calculate 1*2^3 + 1*2^2 + 0*2^1 + 1*2^0 = 8 + 4 + 0 + 1 = 13.
Correct Answer:
A
— 13
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Q. If the decimal number 10 is converted to binary, what is the resulting binary number? (2023)
A.
1010
B.
1001
C.
1100
D.
1110
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Solution
The decimal number 10 is represented as 1010 in binary.
Correct Answer:
A
— 1010
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Q. If the decimal number 15 is converted to binary, what is the resulting binary number?
A.
1110
B.
1111
C.
1101
D.
1011
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Solution
The decimal number 15 is represented as 1111 in binary.
Correct Answer:
B
— 1111
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Q. If the decimal number 255 is converted to binary, what is the resulting binary number?
A.
11111111
B.
11011111
C.
10111111
D.
10011111
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Solution
The decimal number 255 converts to binary as 11111111.
Correct Answer:
A
— 11111111
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Q. If the GCD of two numbers is 1, what can be inferred about the two numbers? (2023)
A.
They are both even.
B.
They are both odd.
C.
They are coprime.
D.
They are multiples of each other.
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Solution
If the GCD of two numbers is 1, it means they have no common factors other than 1, thus they are coprime.
Correct Answer:
C
— They are coprime.
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Q. If the GCD of two numbers is 1, which of the following statements is true?
A.
The numbers are multiples of each other
B.
The numbers are co-prime
C.
The numbers are both even
D.
The numbers are both odd
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Solution
If the GCD of two numbers is 1, it means they have no common factors other than 1, hence they are co-prime.
Correct Answer:
B
— The numbers are co-prime
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Q. If the HCF of 18 and 24 is 6, what is the LCM of these two numbers? (2023)
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Solution
Using the relation HCF * LCM = Product of the numbers, we have 6 * LCM = 18 * 24. Thus, LCM = (18 * 24) / 6 = 72.
Correct Answer:
A
— 72
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Q. If the HCF of 24 and 36 is 12, what is the LCM of these two numbers? (2023)
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Solution
Using the relation HCF * LCM = Product of the numbers, we have LCM = (24 * 36) / 12 = 72.
Correct Answer:
A
— 72
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Q. If the HCF of three numbers is 1, which of the following can be true?
A.
All three numbers are prime
B.
All three numbers are even
C.
All three numbers are odd
D.
Two numbers are even and one is odd
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Solution
If the HCF is 1, it is possible that all three numbers are prime, as they would not share any common factors.
Correct Answer:
A
— All three numbers are prime
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Q. If the HCF of three numbers is 5 and their LCM is 300, what can be said about the numbers?
A.
They are all multiples of 5
B.
They are all prime
C.
They are all even
D.
They are all odd
Show solution
Solution
Since the HCF is 5, all three numbers must be multiples of 5.
Correct Answer:
A
— They are all multiples of 5
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Q. If the HCF of three numbers is 5 and their LCM is 300, what is the product of the three numbers? (2023)
A.
1500
B.
3000
C.
6000
D.
7500
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Solution
The product of the three numbers is equal to HCF * LCM = 5 * 300 = 1500.
Correct Answer:
B
— 3000
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Q. If the HCF of three numbers is 7 and their LCM is 420, what is the maximum possible value of the product of these three numbers? (2023)
A.
2940
B.
840
C.
1260
D.
2100
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Solution
The product of the three numbers is equal to HCF * LCM. Thus, 7 * 420 = 2940.
Correct Answer:
C
— 1260
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Q. If the HCF of three numbers is 7 and their LCM is 420, what is the product of the three numbers? (2023)
A.
2940
B.
840
C.
2100
D.
1470
Show solution
Solution
The product of the three numbers = HCF * LCM = 7 * 420 = 2940.
Correct Answer:
A
— 2940
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Q. If the HCF of two numbers is 1, what can be inferred about the two numbers? (2023)
A.
They are both prime.
B.
They are coprime.
C.
They are multiples of each other.
D.
They are both even.
Show solution
Solution
If the HCF is 1, it means the two numbers are coprime, meaning they have no common factors other than 1.
Correct Answer:
B
— They are coprime.
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Q. If the HCF of two numbers is 1, what can be said about the two numbers? (2023)
A.
They are prime
B.
They are coprime
C.
They are even
D.
They are odd
Show solution
Solution
If the HCF is 1, the two numbers are coprime, meaning they have no common factors other than 1.
Correct Answer:
B
— They are coprime
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Q. If the HCF of two numbers is 12 and their LCM is 120, what is the product of the two numbers?
A.
144
B.
1200
C.
240
D.
100
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Solution
The product of two numbers is equal to the product of their HCF and LCM. Therefore, the product = 12 * 120 = 1440.
Correct Answer:
B
— 1200
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Q. If the HCF of two numbers is 12 and their LCM is 144, what is the product of the two numbers? (2023)
A.
1728
B.
864
C.
144
D.
288
Show solution
Solution
The product of two numbers is equal to the product of their HCF and LCM. Therefore, 12 * 144 = 1728.
Correct Answer:
A
— 1728
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Q. If the HCF of two numbers is 5 and their LCM is 100, what is the sum of the two numbers? (2023)
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Solution
Let the two numbers be 5x and 5y. Then, HCF(5x, 5y) = 5 and LCM(5x, 5y) = 100. This gives xy = 20. The pairs (x, y) that satisfy this are (4, 5) or (5, 4), leading to the sum 5(4 + 5) = 45.
Correct Answer:
A
— 30
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Q. If the HCF of two numbers is 5 and their LCM is 100, what is the sum of the two numbers if one of them is 25?
A.
50
B.
75
C.
100
D.
125
Show solution
Solution
Let the other number be x. Then, HCF(25, x) = 5 and LCM(25, x) = 100. Therefore, 25 * x = 5 * 100, which gives x = 20. The sum is 25 + 20 = 45.
Correct Answer:
A
— 50
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Q. If the HCF of two numbers is 5 and their product is 100, what could be the two numbers? (2023)
A.
5 and 20
B.
10 and 10
C.
15 and 5
D.
25 and 4
Show solution
Solution
If HCF is 5, the two numbers can be expressed as 5a and 5b. Thus, 5a * 5b = 100 implies ab = 4. The pair (1, 4) gives us 5 and 20.
Correct Answer:
A
— 5 and 20
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Q. If the HCF of two numbers is 5 and their product is 100, what is their LCM?
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Solution
Using the relationship between HCF, LCM, and product: HCF * LCM = Product. Thus, LCM = 100 / 5 = 20.
Correct Answer:
B
— 25
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Q. If the HCF of two numbers is 5 and their product is 1000, what is their LCM? (2023)
A.
200
B.
100
C.
250
D.
150
Show solution
Solution
Using the relation HCF * LCM = Product of the numbers, we have LCM = 1000 / 5 = 200.
Correct Answer:
A
— 200
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Q. If the HCF of two numbers is 8 and their LCM is 48, what could be one of the possible pairs of numbers? (2023)
A.
16 and 24
B.
8 and 32
C.
4 and 12
D.
20 and 16
Show solution
Solution
The numbers can be 16 and 24, as their HCF is 8 and LCM is 48.
Correct Answer:
A
— 16 and 24
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Q. If the HCF of two numbers is 8 and their product is 288, what is their LCM? (2023)
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Solution
Using the relationship between HCF, LCM, and the product of two numbers: LCM = Product / HCF = 288 / 8 = 36.
Correct Answer:
A
— 36
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Q. If the hexadecimal number A3 is converted to decimal, what is its value? (2023)
A.
163
B.
1630
C.
103
D.
83
Show solution
Solution
A3 in hexadecimal is calculated as 10*16^1 + 3*16^0 = 160 + 3 = 163.
Correct Answer:
A
— 163
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Showing 181 to 210 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!
