Q. A gardener has 36 red roses and 48 yellow roses. He wants to plant them in rows with the same number of each type of rose in each row. What is the maximum number of rows he can plant? (2023)
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Solution
The HCF of 36 and 48 is 12, which is the maximum number of rows he can plant.
Correct Answer: B — 12
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Q. A gardener has two types of plants, one type has a height of 3 feet and the other 5 feet. What is the minimum height at which both types can be tied together? (2023)
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Solution
The minimum height is the LCM of 3 and 5, which is 15 feet.
Correct Answer: C — 60
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Q. A number is divided by 11 and gives a remainder of 4. If this number is multiplied by 3, what will be the remainder when the result is divided by 11?
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Solution
The new number is 3*(11k + 4) = 33k + 12, and 12 mod 11 = 1.
Correct Answer: B — 2
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Q. A number is divided by 7 and gives a remainder of 3. If this number is increased by 4, what will be the new remainder when divided by 7?
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Solution
The new number is (7k + 3 + 4) = 7k + 7, which gives a remainder of 0 when divided by 7.
Correct Answer: A — 0
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Q. A number is divisible by both 12 and 15. What is the smallest number that is also divisible by 36? (2023)
A.
180
B.
120
C.
240
D.
360
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Solution
The LCM of 12 and 15 is 60. The smallest number divisible by 60 and 36 is 180.
Correct Answer: A — 180
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Q. A number leaves a remainder of 1 when divided by 5 and a remainder of 2 when divided by 7. What is the smallest such number?
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Solution
The smallest number satisfying both conditions is 22.
Correct Answer: C — 22
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Q. A teacher has 48 pencils and 60 erasers. She wants to distribute them equally among students. What is the maximum number of students she can distribute to? (2023)
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Solution
The HCF of 48 and 60 is 12, which is the maximum number of students she can distribute to equally.
Correct Answer: A — 12
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Q. For a number to be divisible by 10, which of the following must be true?
A.
It must end in 0
B.
It must be a two-digit number
C.
It must be a prime number
D.
It must be even
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Solution
A number is divisible by 10 if it ends in 0.
Correct Answer: A — It must end in 0
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Q. How does the author support the claims made about digital sum? (2023)
A.
By providing historical examples.
B.
By citing recent technological advancements.
C.
By referencing mathematical theories.
D.
By discussing personal experiences.
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Solution
The author supports the claims about digital sum by citing recent technological advancements that utilize this concept.
Correct Answer: B — By citing recent technological advancements.
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Q. If 'A' is a number in base-5 and is represented as '43', what is its decimal equivalent?
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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 'XYZ' in base-4 equals 27 in decimal, what is the value of 'X'?
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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 4x ≡ 8 (mod 12), what is the smallest non-negative integer solution for x?
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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 7x ≡ 3 (mod 5), what is the value of x?
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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?
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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-3 is represented as '210', what is its decimal equivalent?
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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?
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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 is a factor of 36, which of the following numbers is NOT a factor?
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Solution
All options are factors of 36 except for 9, which is a factor but not a multiple of 4.
Correct Answer: D — 9
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Q. If a number is a multiple of 9, which of the following must also be true?
A.
It is a multiple of 3.
B.
It is a multiple of 27.
C.
It is a prime number.
D.
It is an even number.
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Solution
All multiples of 9 are also multiples of 3.
Correct Answer: A — It is a multiple of 3.
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Q. If a number is a multiple of both 4 and 6, which of the following must it also be a multiple of?
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Solution
The least common multiple of 4 and 6 is 12, so any multiple of both must also be a multiple of 12.
Correct Answer: A — 12
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Q. If a number is a multiple of both 7 and 9, which of the following numbers is a possible candidate?
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Solution
63 is a multiple of both 7 (7x9) and 9 (9x7).
Correct Answer: A — 63
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Q. If a number is congruent to 0 modulo 6, which of the following could be a possible value?
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Solution
A number congruent to 0 mod 6 must be a multiple of 6. 12 is a multiple of 6.
Correct Answer: B — 12
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Q. If a number is divided by 9 and gives a remainder of 2, which of the following numbers will also give the same remainder when divided by 9?
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Solution
29 gives a remainder of 2 when divided by 9 (9*3 + 2 = 29).
Correct Answer: C — 29
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Q. If a number is divisible by 12, which of the following must also be true?
A.
It is divisible by 3
B.
It is divisible by 5
C.
It is divisible by 10
D.
It is a prime number
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Solution
A number divisible by 12 is also divisible by both 3 and 4.
Correct Answer: A — It is divisible by 3
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Q. If a number is divisible by 4, which of the following must also be true?
A.
It is even
B.
It is divisible by 8
C.
It is divisible by 2
D.
It is a multiple of 10
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Solution
Any number divisible by 4 is even, as 4 itself is an even number.
Correct Answer: A — It is even
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Q. If a number is divisible by 7, which of the following is NOT necessarily true?
A.
It is odd
B.
It is not a prime number
C.
It can be a multiple of 14
D.
It can be a two-digit number
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Solution
A number can be divisible by 7 and still be even, such as 14.
Correct Answer: A — It is odd
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Q. If a number is divisible by 8, which of the following must also be true?
A.
It is divisible by 2
B.
It is divisible by 4
C.
It is a multiple of 16
D.
It is a prime number
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Solution
Any number divisible by 8 is also divisible by 4, as 8 is a multiple of 4.
Correct Answer: B — It is divisible by 4
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Q. If a number is divisible by both 2 and 3, which of the following must it also be divisible by?
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Solution
A number divisible by both 2 and 3 is also divisible by their least common multiple, which is 6.
Correct Answer: B — 6
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Q. If a number is divisible by both 3 and 5, which of the following must also be true?
A.
It is divisible by 15
B.
It is divisible by 8
C.
It is divisible by 10
D.
It is not divisible by 6
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Solution
A number divisible by both 3 and 5 is also divisible by their LCM, which is 15.
Correct Answer: A — It is divisible by 15
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Q. If a number is divisible by both 8 and 12, which of the following must it also be divisible by?
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Solution
The least common multiple of 8 and 12 is 24, so the number must be divisible by 24.
Correct Answer: B — 24
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Q. If a number is expressed as 5k + 3, where k is an integer, what is the remainder when this number is divided by 5?
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Solution
The remainder is 3, as 5k is divisible by 5.
Correct Answer: D — 3
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