Number Systems for Competitive Exam Success

Number Systems

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

Q. If 'A' in hexadecimal is equal to 10 in decimal, what is 'A + 5' in hexadecimal?

A. F
B. G
C. B
D. C

Solution

'A' (10) + 5 = 15, which is 'F' in hexadecimal.

Correct Answer: A — F

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Q. If 'A' is a number in base-4 that equals 10 in decimal, what is 'A'?

A. 22
B. 24
C. 30
D. 32

Solution

In base-4, '22' equals 2*4^1 + 2*4^0 = 8 + 2 = 10.

Correct Answer: A — 22

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Q. If 'A' is a number in base-5 and is represented as '43', what is its decimal equivalent?

A. 20
B. 23
C. 25
D. 30

Solution

'43' in base 5 is calculated as 4*5^1 + 3*5^0 = 20 + 3 = 23.

Correct Answer: B — 23

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Q. If 'A' represents 10 in a hexadecimal system, what is the decimal equivalent of '1A'?

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

Solution

'1A' in hexadecimal is calculated as 1*16^1 + 10*16^0 = 16 + 10 = 26.

Correct Answer: A — 26

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Q. If 'C' in hexadecimal is equal to 12 in decimal, what is the value of 'D'?

A. 11
B. 12
C. 13
D. 14

Solution

'D' in hexadecimal represents 13 in decimal.

Correct Answer: C — 13

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Q. If 'C' represents 12 in a hexadecimal system, what is the decimal equivalent of '2C'?

A. 44
B. 45
C. 46
D. 47

Solution

'2C' in hexadecimal is calculated as 2*16^1 + 12*16^0 = 32 + 12 = 44.

Correct Answer: A — 44

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Q. If 'x' in base-7 is equal to 50 in decimal, what is 'x'?

A. 101
B. 110
C. 120
D. 130

Solution

50 in decimal is '110' in base-7 (1*7^2 + 1*7^1 + 0*7^0 = 50).

Correct Answer: B — 110

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Q. If 'x' is a number in base 5 represented as '320', what is its decimal equivalent?

A. 75
B. 80
C. 85
D. 90

Solution

'320' in base 5 = 3*5^2 + 2*5^1 + 0*5^0 = 75.

Correct Answer: A — 75

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Q. If 'XYZ' in a base-6 system equals 215 in decimal, what is the value of 'X'?

A. 3
B. 4
C. 5
D. 2

Solution

215 in decimal is '543' in base-6, so 'X' is 4.

Correct Answer: B — 4

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Q. If 'XYZ' in base 5 equals 100 in decimal, what is the value of 'X'?

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

Solution

'XYZ' in base 5 means X*25 + Y*5 + Z = 100. The only valid solution is X=2.

Correct Answer: B — 2

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Q. If 'XYZ' in base-3 equals 26 in decimal, what is the value of 'X'?

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

Solution

In base-3, 'XYZ' means X*9 + Y*3 + Z = 26. The only valid digit for X is 2.

Correct Answer: A — 2

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Q. If 'XYZ' in base-4 equals 27 in decimal, what is the value of 'X'?

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

Solution

In base-4, 'XYZ' = 4^2*X + 4^1*Y + 4^0*Z = 27. The only valid combination is X=3, Y=3, Z=3.

Correct Answer: C — 3

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Q. If 15 is congruent to x modulo 6, what is the value of x?

A. 3
B. 9
C. 15
D. 21

Solution

15 mod 6 = 3, so x = 3.

Correct Answer: A — 3

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Q. If 2x ≡ 4 (mod 6), what is the smallest non-negative integer solution for x?

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

Solution

Dividing both sides by 2 gives x ≡ 2 (mod 3), hence the smallest non-negative solution is 2.

Correct Answer: C — 2

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Q. If 3x ≡ 9 (mod 12), what is the value of x?

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

Solution

Dividing both sides by 3 gives x ≡ 3 (mod 12), which means x can be 3.

Correct Answer: B — 2

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Q. If 3x ≡ 9 (mod 6), what is the value of x?

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

Solution

3x = 9 mod 6 simplifies to x = 3 mod 2, so x = 2.

Correct Answer: C — 2

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Q. If 4x ≡ 1 (mod 9), what is the smallest positive integer solution for x?

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

Solution

The multiplicative inverse of 4 mod 9 is 7, since 4 * 7 = 28 ≡ 1 (mod 9).

Correct Answer: A — 1

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Q. If 4x ≡ 8 (mod 12), what is the smallest non-negative integer solution for x?

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

Solution

Dividing both sides by 4 gives x ≡ 2 (mod 3), so the smallest non-negative solution is 2.

Correct Answer: C — 2

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Q. If 4x ≡ 8 (mod 12), what is the smallest non-negative solution for x?

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

Solution

Dividing the equation by 4 gives x ≡ 2 (mod 3). The smallest non-negative solution is x = 2.

Correct Answer: C — 2

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Q. If 5x ≡ 10 (mod 15), what is the value of x?

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

Solution

Dividing both sides by 5 gives x ≡ 2 (mod 3), which means x can be 2.

Correct Answer: C — 2

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Q. If 7x ≡ 3 (mod 5), what is the value of x?

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

Solution

To solve 7x ≡ 3 (mod 5), we first reduce 7 mod 5 to get 2x ≡ 3 (mod 5). The solution is x ≡ 4 (mod 5), which corresponds to 2.

Correct Answer: C — 3

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Q. If 8x ≡ 4 (mod 12), what is the value of x?

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

Solution

Dividing both sides by 4 gives 2x ≡ 1 (mod 3). The solution is x = 2.

Correct Answer: B — 2

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Q. If a number in base 4 is represented as '210', what is its equivalent in base 10?

A. 32
B. 34
C. 36
D. 38

Solution

'210' in base 4 = 2*4^2 + 1*4^1 + 0*4^0 = 32.

Correct Answer: A — 32

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Q. If a number in base 5 is represented as '243', what is its equivalent in decimal?

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

Solution

'243' in base 5 is calculated as 2*5^2 + 4*5^1 + 3*5^0 = 50 + 20 + 3 = 73.

Correct Answer: C — 62

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Q. If a number in base 5 is represented as '320', what is its decimal equivalent? (2023)

A. 75
B. 80
C. 50
D. 25

Solution

The decimal equivalent of '320' in base 5 is calculated as (3 * 5^2) + (2 * 5^1) + (0 * 5^0) = 75 + 10 + 0 = 80.

Correct Answer: B — 80

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Q. If a number in base 5 is represented as 243, what is its decimal equivalent? (2023)

A. 60
B. 55
C. 50
D. 45

Solution

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

Correct Answer: A — 60

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Q. If a number in base 8 is represented as '57', what is its equivalent in base 10?

A. 47
B. 49
C. 50
D. 51

Solution

'57' in base 8 = 5*8^1 + 7*8^0 = 40 + 7 = 47 in decimal.

Correct Answer: B — 49

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Q. If a number in base-3 is represented as '210', what is its decimal equivalent?

A. 18
B. 19
C. 20
D. 21

Solution

'210' in base-3 is 2*3^2 + 1*3^1 + 0*3^0 = 18 + 3 + 0 = 21.

Correct Answer: B — 19

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Q. If a number in base-6 is represented as '54', what is its decimal equivalent?

A. 30
B. 32
C. 34
D. 36

Solution

'54' in base 6 is calculated as 5*6^1 + 4*6^0 = 30 + 4 = 34.

Correct Answer: B — 32

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Q. If a number in base-6 is represented as 405, what is its decimal equivalent?

A. 145
B. 150
C. 155
D. 160

Solution

To convert from base-6 to decimal, calculate 4*6^2 + 0*6^1 + 5*6^0 = 4*36 + 0 + 5 = 144 + 5 = 149.

Correct Answer: B — 150

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Number Systems

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