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 '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 '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 '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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Number Systems

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