Q. If a number leaves a remainder of 1 when divided by 4 and a remainder of 2 when divided by 5, what is the smallest positive integer that satisfies these conditions?
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Solution
The smallest number satisfying both conditions is 9, as 9 mod 4 = 1 and 9 mod 5 = 4.
Correct Answer: A — 6
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Q. If a number leaves a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7, what is the smallest such number?
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Solution
The smallest number satisfying both conditions is 12, as 12 mod 5 = 2 and 12 mod 7 = 5.
Correct Answer: A — 12
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Q. If a number leaves a remainder of 4 when divided by 8, which of the following numbers will also leave the same remainder when divided by 8?
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Solution
20 leaves a remainder of 4 when divided by 8 (8*2 + 4 = 20).
Correct Answer: B — 20
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Q. If a number leaves a remainder of 5 when divided by 12, what will be the remainder when the same number is divided by 4?
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Solution
The number can be expressed as 12k + 5. When divided by 4, the remainder is 5 mod 4, which is 1.
Correct Answer: B — 2
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Q. If a number leaves a remainder of 5 when divided by 12, which of the following could be the number?
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Solution
17 leaves a remainder of 5 when divided by 12 (12*1 + 5 = 17).
Correct Answer: A — 17
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Q. If a number leaves a remainder of 7 when divided by 10, what will be the remainder when the same number is divided by 5?
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Solution
The number can be expressed as 10k + 7. When divided by 5, the remainder is 7 mod 5, which is 2.
Correct Answer: C — 2
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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 ≡ 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 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 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 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 LCM of three numbers is 180 and their HCF is 3, what is the product of the three numbers? (2023)
A.
540
B.
1800
C.
5400
D.
18000
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Solution
The product of the three numbers is equal to LCM * HCF^2. Therefore, 180 * 3^2 = 180 * 9 = 1620.
Correct Answer: C — 5400
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Q. If the LCM of two numbers is 120 and their HCF is 10, what is the product of the two numbers? (2023)
A.
1200
B.
1000
C.
2400
D.
600
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Solution
The product of two numbers is equal to the product of their LCM and HCF. Therefore, 120 * 10 = 1200.
Correct Answer: A — 1200
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Q. If the LCM of two numbers is 60 and their HCF is 4, what is the product of the two numbers? (2023)
A.
240
B.
120
C.
300
D.
180
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Solution
The product of two numbers is equal to the product of their LCM and HCF. Therefore, 60 * 4 = 240.
Correct Answer: A — 240
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Q. If the least common multiple (LCM) of two numbers is 60 and their greatest common divisor (GCD) is 12, what is the product of the two numbers?
A.
720
B.
180
C.
120
D.
60
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Solution
The product of two numbers is equal to the product of their LCM and GCD. Therefore, 60 * 12 = 720.
Correct Answer: A — 720
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Q. If the product of two consecutive integers is 72, which of the following pairs could represent these integers?
A.
8 and 9
B.
6 and 7
C.
5 and 6
D.
9 and 10
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Solution
6 and 7 are consecutive integers whose product is 42, not 72.
Correct Answer: B — 6 and 7
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Q. If the product of two numbers is a multiple of 15, which of the following must be true?
A.
At least one of the numbers is a multiple of 3.
B.
At least one of the numbers is a multiple of 5.
C.
Both numbers are even.
D.
Both numbers are odd.
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Solution
For the product to be a multiple of 15, at least one of the numbers must be a multiple of 5.
Correct Answer: B — At least one of the numbers is a multiple of 5.
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Q. If the sum of two multiples of 7 is 56, which of the following pairs could represent these multiples?
A.
21 and 35
B.
14 and 42
C.
28 and 28
D.
All of the above.
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Solution
All pairs listed are multiples of 7 that sum to 56.
Correct Answer: D — All of the above.
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Q. If x ≡ 3 (mod 7) and x ≡ 5 (mod 11), what is the smallest positive integer solution for x?
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Solution
Using the method of successive substitutions or the Chinese Remainder Theorem, the smallest solution is x = 26.
Correct Answer: B — 26
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Q. If x ≡ 4 (mod 7) and y ≡ 3 (mod 7), what is the value of (x + y) mod 7?
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Solution
x + y ≡ 4 + 3 ≡ 7 (mod 7), which is equivalent to 0.
Correct Answer: A — 0
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Q. In a base-3 number system, what is the product of '10' and '11'?
A.
100
B.
110
C.
101
D.
111
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Solution
'10' in base 3 is 3 and '11' is 4. Their product is 12, which is '110' in base 3.
Correct Answer: A — 100
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Q. In a base-4 number system, what is the decimal equivalent of the number '132'?
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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-5 number system, what is the sum of '34' and '21'?
A.
100
B.
110
C.
120
D.
130
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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 number 243?
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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, which of the following digits is not valid?
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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'?
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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 certain division problem, if the dividend is 123 and the divisor is 10, what is the remainder?
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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 number system, the number '101' represents the decimal number 5. What does '111' represent in the same system?
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Solution
'101' in binary is 5, so '111' in binary is 7.
Correct Answer: B — 7
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