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
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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
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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
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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
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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
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Solution
A3 in hexadecimal is calculated as 10*16^1 + 3*16^0 = 160 + 3 = 163.
Correct Answer:
A
— 163
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