Number Systems for Competitive Exam Success

Number Systems

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

Q. In a modular arithmetic system, if 7 is congruent to x modulo 5, what is the value of x?

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

Solution

To find x, we calculate 7 mod 5, which is 2. Therefore, x = 2.

Correct Answer: A — 2

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Q. In a modular arithmetic system, if 8 is congruent to 2 modulo n, what can be concluded about n?

A. n must be 6
B. n must be a factor of 6
C. n must be greater than 6
D. n must be less than 6

Solution

Since 8 - 2 = 6, n must be a divisor of 6.

Correct Answer: B — n must be a factor of 6

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Q. In a modular arithmetic system, if 9 is congruent to y modulo 4, what is the value of y?

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

Solution

9 mod 4 = 1, so y = 1.

Correct Answer: A — 1

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Q. In a modular arithmetic system, if a ≡ b (mod m) and c ≡ d (mod m), which of the following is not necessarily true?

A. a + c ≡ b + d (mod m)
B. a - c ≡ b - d (mod m)
C. a * c ≡ b * d (mod m)
D. a / c ≡ b / d (mod m)

Solution

Division is not guaranteed in modular arithmetic, as it requires the divisor to have a multiplicative inverse.

Correct Answer: D — a / c ≡ b / d (mod m)

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Q. In a modular arithmetic system, if a ≡ b (mod m), which of the following statements is true?

A. a - b is divisible by m
B. a + b is divisible by m
C. a * b is divisible by m
D. a / b is divisible by m

Solution

The statement a ≡ b (mod m) means that the difference a - b is divisible by m.

Correct Answer: A — a - b is divisible by m

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Q. In a modular system with modulus 12, what is the result of 15 + 10?

A. 5
B. 7
C. 3
D. 1

Solution

15 + 10 = 25; 25 mod 12 = 1.

Correct Answer: A — 5

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Q. In a modular system with modulus 5, what is the result of (3 + 4) mod 5?

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

Solution

(3 + 4) = 7, and 7 mod 5 = 2.

Correct Answer: C — 0

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Q. In a modular system, if 12 is congruent to 0 modulo n, which of the following must be true?

A. n is a factor of 12
B. n is greater than 12
C. n is less than 12
D. n is a prime number

Solution

For 12 to be congruent to 0 modulo n, n must be a divisor of 12.

Correct Answer: A — n is a factor of 12

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Q. In a modular system, 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

Solution

All operations (addition, subtraction, multiplication) maintain equivalence in modular arithmetic.

Correct Answer: D — All of the above

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Q. In a modular system, if a ≡ b (mod n) and c ≡ d (mod n), which of the following is true?

A. a + c ≡ b + d (mod n)
B. a - c ≡ b - d (mod n)
C. a * c ≡ b * d (mod n)
D. All of the above

Solution

All operations maintain the equivalence in modular arithmetic.

Correct Answer: D — All of the above

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Q. In a number system where '2' represents 10, what is the value of '4'?

A. 20
B. 30
C. 40
D. 50

Solution

'4' in this system represents 20 in decimal since '2' is 10.

Correct Answer: B — 30

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Q. In a number system where 'A' represents 10, 'B' represents 11, and 'C' represents 12, what is the decimal equivalent of 'BAC'?

A. 186
B. 187
C. 188
D. 189

Solution

'BAC' = 11*16^2 + 10*16^1 + 12*16^0 = 186 + 10 + 12 = 188.

Correct Answer: B — 187

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Q. In a number system where 'A' represents 10, 'B' represents 11, and 'C' represents 12, what is the decimal equivalent of 'BAA'?

A. 121
B. 122
C. 123
D. 124

Solution

'BAA' = 11*100 + 10*10 + 10 = 121 + 20 + 10 = 121.

Correct Answer: C — 123

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Q. In a number system where 'B' represents 11, what is the value of 'B2' in decimal?

A. 12
B. 22
C. 21
D. 20

Solution

'B2' = 11*5^1 + 2*5^0 = 55 + 2 = 57 in decimal.

Correct Answer: C — 21

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Q. In a number system, if '100' represents 4, what does '110' represent?

A. 5
B. 6
C. 7
D. 8

Solution

'100' in base 4 is 4. '110' in base 4 is 1*16 + 1*4 + 0 = 20, which is 6.

Correct Answer: B — 6

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Q. In a number system, if 'C' represents 12, what is the value of 'C0' in decimal?

A. 120
B. 121
C. 122
D. 123

Solution

'C0' = 12*10^1 + 0*10^0 = 120 in decimal.

Correct Answer: A — 120

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Q. In a number system, if the base is increased, what happens to the number of unique digits available?

A. It decreases.
B. It remains the same.
C. It increases.
D. It becomes zero.

Solution

Increasing the base of a number system increases the number of unique digits available for representation.

Correct Answer: C — It increases.

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Q. In a numeral system where the base is 6, what is the value of '34'?

A. 20
B. 22
C. 24
D. 26

Solution

'34' in base 6 is calculated as 3*6^1 + 4*6^0 = 18 + 4 = 22.

Correct Answer: C — 24

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Q. In a numeral system where the digits are 0, 1, 2, 3, and 4, what is the value of '34' in decimal?

A. 12
B. 14
C. 16
D. 18

Solution

'34' in base 5 is calculated as 3*5^1 + 4*5^0 = 15 + 4 = 19.

Correct Answer: C — 16

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Q. In a numeral system, what does the term 'base' refer to? (2023)

A. The number of unique digits used.
B. The maximum value of a digit.
C. The total number of digits in a number.
D. The position of a digit in a number.

Solution

The term 'base' refers to the number of unique digits used in a numeral system.

Correct Answer: A — The number of unique digits used.

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Q. In a numeral system, what does the term 'place value' refer to?

A. The value of a digit based on its position.
B. The total value of all digits combined.
C. The maximum value a numeral can represent.
D. The number of digits in a numeral.

Solution

Place value refers to the value of a digit based on its position in a numeral.

Correct Answer: A — The value of a digit based on its position.

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Q. In a sequence of numbers where each number is a multiple of 3, which of the following cannot be a member of this sequence?

A. 9
B. 15
C. 22
D. 27

Solution

22 is not a multiple of 3, while 9, 15, and 27 are.

Correct Answer: C — 22

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Q. In a sequence of numbers where each number is a multiple of 7, which of the following could be the 5th number in the sequence?

A. 28
B. 35
C. 42
D. 49

Solution

The 5th number in the sequence of multiples of 7 is 7 * 5 = 35, so 42 is the next multiple after 35.

Correct Answer: C — 42

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Q. In a sequence of numbers where each number is a multiple of 8, which of the following could be the 5th number in the sequence?

A. 32
B. 40
C. 56
D. 72

Solution

The 5th number in a sequence of multiples of 8 could be 32 (8 x 4), while the others are not multiples of 8.

Correct Answer: A — 32

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Q. In a sequence of numbers where each number is a multiple of 9, which of the following numbers cannot be in the sequence?

A. 27
B. 45
C. 81
D. 50

Solution

50 is not a multiple of 9, while 27, 45, and 81 are all multiples of 9.

Correct Answer: D — 50

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Q. In a set of numbers, if the number 30 is a multiple of a certain number 'X', which of the following could be 'X'?

A. 5
B. 7
C. 10
D. 12

Solution

Both 5 and 10 are factors of 30, making them valid options for 'X'.

Correct Answer: A — 5

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Q. In converting the decimal number 255 to hexadecimal, what is the result?

A. FF
B. F0
C. 00
D. 255

Solution

The decimal number 255 is represented as FF in hexadecimal.

Correct Answer: A — FF

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Q. In converting the hexadecimal number 'A3' to decimal, what is the resulting value?

A. 163
B. 1632
C. 123
D. 103

Solution

The hexadecimal number 'A3' converts to decimal as follows: A (10)*16^1 + 3*16^0 = 160 + 3 = 163.

Correct Answer: A — 163

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Q. In converting the hexadecimal number 1A to decimal, what is the result?

A. 26
B. 27
C. 28
D. 29

Solution

The hexadecimal number 1A converts to decimal as 1*16^1 + 10*16^0 = 26.

Correct Answer: A — 26

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Q. In modular arithmetic, what is the multiplicative inverse of 3 mod 11?

A. 4
B. 7
C. 8
D. 9

Solution

The multiplicative inverse of 3 mod 11 is 4, since 3 * 4 ≡ 12 ≡ 1 (mod 11).

Correct Answer: A — 4

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Number Systems

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