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?
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?
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A.
a + c ≡ b + d (mod m)
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B.
a - c ≡ b - d (mod m)
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C.
a * c ≡ b * d (mod m)
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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
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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?
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?
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
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B.
n is greater than 12
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C.
n is less than 12
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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 modular arithmetic, what is the multiplicative inverse of 3 mod 11?
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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Q. In modular arithmetic, what is the multiplicative inverse of 3 modulo 11?
Solution
The multiplicative inverse of 3 mod 11 is 4, since (3 * 4) mod 11 = 12 mod 11 = 1.
Correct Answer:
B
— 7
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Q. In modular arithmetic, which of the following is a valid operation?
-
A.
Adding two numbers and taking mod
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B.
Subtracting two numbers and taking mod
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C.
Multiplying two numbers and taking mod
-
D.
All of the above
Solution
All operations (addition, subtraction, multiplication) are valid in modular arithmetic.
Correct Answer:
D
— All of the above
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Q. In modular arithmetic, which of the following is true for any integer a?
-
A.
a mod 1 = 0
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B.
a mod a = 1
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C.
a mod 0 is undefined
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D.
a mod 2 = 0 or 1
Solution
For any integer a, a mod 2 will always yield either 0 or 1, depending on whether a is even or odd.
Correct Answer:
D
— a mod 2 = 0 or 1
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Q. In modular arithmetic, which of the following is true for any integer k?
-
A.
k mod 1 = 0
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B.
k mod k = 1
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C.
k mod 0 is undefined
-
D.
k mod k = 0
Solution
For any integer k, k mod k = 0, as k is divisible by itself.
Correct Answer:
D
— k mod k = 0
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Q. What is the modular inverse of 3 modulo 11?
Solution
The modular inverse of 3 mod 11 is a number x such that 3x ≡ 1 (mod 11). The solution is x = 4, since 3 * 4 = 12 ≡ 1 (mod 11).
Correct Answer:
A
— 4
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Q. What is the multiplicative inverse of 3 modulo 11?
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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Q. What is the multiplicative inverse of 3 modulo 7?
Solution
The multiplicative inverse of 3 mod 7 is 5 because (3 * 5) mod 7 = 15 mod 7 = 1.
Correct Answer:
A
— 2
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Q. What is the result of (10 + 15) mod 7?
Solution
(10 + 15) = 25; 25 mod 7 = 4.
Correct Answer:
B
— 2
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Q. What is the result of (15 + 10) mod 12?
Solution
(15 + 10) = 25; 25 mod 12 = 1.
Correct Answer:
A
— 5
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Q. What is the result of (3 * 4) mod 5?
Solution
Calculating (3 * 4) = 12, and then 12 mod 5 = 2.
Correct Answer:
C
— 4
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Q. What is the result of 10 mod 3?
Solution
10 divided by 3 gives a remainder of 1, so 10 mod 3 = 1.
Correct Answer:
B
— 2
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Q. Which of the following equations has no solution in modular arithmetic?
-
A.
2x ≡ 4 (mod 6)
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B.
3x ≡ 9 (mod 6)
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C.
5x ≡ 10 (mod 6)
-
D.
4x ≡ 8 (mod 6)
Solution
3x ≡ 9 (mod 6) has no solution because 3 and 6 are not coprime, and 9 is not divisible by the gcd(3, 6).
Correct Answer:
B
— 3x ≡ 9 (mod 6)
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Q. Which of the following equations is true in modular arithmetic?
-
A.
5 ≡ 10 (mod 5)
-
B.
6 ≡ 12 (mod 6)
-
C.
7 ≡ 14 (mod 7)
-
D.
8 ≡ 15 (mod 8)
Solution
8 ≡ 15 (mod 8) is false; the correct answer is 5 ≡ 10 (mod 5) is true.
Correct Answer:
D
— 8 ≡ 15 (mod 8)
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Q. Which of the following equations represents a valid modular arithmetic statement?
-
A.
5x ≡ 10 (mod 5)
-
B.
6x ≡ 12 (mod 6)
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C.
7x ≡ 14 (mod 7)
-
D.
8x ≡ 16 (mod 8)
Solution
All options are valid, but 5x ≡ 10 (mod 5) simplifies to 0 ≡ 0, which is trivially true.
Correct Answer:
A
— 5x ≡ 10 (mod 5)
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Q. Which of the following is NOT a property of modular arithmetic?
-
A.
Closure
-
B.
Associativity
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C.
Distributivity
-
D.
Non-commutativity
Solution
Modular arithmetic is commutative for addition and multiplication, hence non-commutativity is not a property.
Correct Answer:
D
— Non-commutativity
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Q. Which of the following is the correct representation of 15 modulo 4?
Solution
15 divided by 4 gives a remainder of 3, hence 15 mod 4 = 3.
Correct Answer:
A
— 3
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Q. Which of the following pairs of numbers are congruent modulo 6?
-
A.
8 and 14
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B.
10 and 16
-
C.
12 and 18
-
D.
All of the above
Solution
All pairs have the same remainder when divided by 6: 8 mod 6 = 2, 14 mod 6 = 2; 10 mod 6 = 4, 16 mod 6 = 4; 12 mod 6 = 0, 18 mod 6 = 0.
Correct Answer:
D
— All of the above
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Q. Which of the following statements is true regarding modular arithmetic?
-
A.
It is only applicable to integers.
-
B.
It can be used for real numbers.
-
C.
It is not useful in computer science.
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D.
It is only used in cryptography.
Solution
Modular arithmetic is primarily applicable to integers, making the first statement true.
Correct Answer:
A
— It is only applicable to integers.
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Q. Which of the following statements is true regarding the expression (a + b) mod n?
-
A.
It is always equal to (a mod n) + (b mod n)
-
B.
It is always equal to (a mod n) - (b mod n)
-
C.
It is always equal to (a mod n) * (b mod n)
-
D.
It is always equal to (a mod n) / (b mod n)
Solution
The property of modular arithmetic states that (a + b) mod n = [(a mod n) + (b mod n)] mod n.
Correct Answer:
A
— It is always equal to (a mod n) + (b mod n)
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