HCF & LCM: Essential Concepts for Competitive Exams

HCF & LCM

Q. A box contains 36 red balls and 48 blue balls. What is the maximum number of equal groups that can be formed with the balls? (2023)

A. 6
B. 12
C. 18
D. 24

Solution

The HCF of 36 and 48 is 12. Therefore, the maximum number of equal groups is 12.

Correct Answer: B — 12

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Q. A factory produces two types of widgets. The first type is produced every 12 days and the second every 15 days. How often will both types be produced on the same day? (2023)

A. 30 days
B. 60 days
C. 45 days
D. 75 days

Solution

The LCM of 12 and 15 is 60, so both types will be produced on the same day every 60 days.

Correct Answer: C — 45 days

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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)

A. 6
B. 12
C. 18
D. 24

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 60 red flowers and 90 yellow flowers. What is the largest number of bouquets he can make if each bouquet has the same number of red and yellow flowers? (2023)

A. 15
B. 30
C. 45
D. 60

Solution

The largest number of bouquets is the HCF of 60 and 90, which is 30.

Correct Answer: B — 30

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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)

A. 15
B. 30
C. 60
D. 45

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 gardener has two types of plants, one type requires watering every 4 days and the other every 6 days. If both types are watered together today, in how many days will they be watered together again? (2023)

A. 12
B. 24
C. 18
D. 30

Solution

The LCM of 4 and 6 is 12. Therefore, they will be watered together again in 12 days.

Correct Answer: A — 12

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Q. A group of students can complete a project in 12 days. If 4 more students join, they can complete it in 8 days. How many students were initially in the group?

A. 6
B. 8
C. 10
D. 12

Solution

Let the initial number of students be x. The work done is inversely proportional to the number of days. Setting up the equation gives x = 10.

Correct Answer: C — 10

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Q. A group of students can complete a project in 12 days. If 4 more students join, the project can be completed in 8 days. How many students were initially in the group? (2023)

A. 6
B. 8
C. 10
D. 12

Solution

Let the initial number of students be x. The work done is constant, so x * 12 = (x + 4) * 8. Solving gives x = 8.

Correct Answer: B — 8

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Q. A number is divisible by both 12 and 15. What is the smallest number that is also divisible by both? (2023)

A. 30
B. 60
C. 90
D. 120

Solution

The LCM of 12 and 15 is 60, which is the smallest number divisible by both.

Correct Answer: B — 60

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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

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 is divisible by both 4 and 6. What is the smallest number that is also divisible by 12?

A. 12
B. 24
C. 36
D. 48

Solution

The smallest number divisible by both 4 and 6 is their LCM, which is 12. Hence, the answer is 12.

Correct Answer: A — 12

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Q. A number is divisible by both 6 and 8. What is the smallest number that is also divisible by 12? (2023)

A. 24
B. 48
C. 36
D. 60

Solution

The LCM of 6 and 8 is 24, which is also divisible by 12. Hence, the smallest number is 24.

Correct Answer: A — 24

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Q. A number is divisible by both 8 and 14. What is the least number of digits in the smallest number that is divisible by both? (2023)

A. 1
B. 2
C. 3
D. 4

Solution

The LCM of 8 and 14 is 56, which is a two-digit number.

Correct Answer: B — 2

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Q. A teacher has 36 pencils and 48 erasers. What is the maximum number of students that can receive the same number of pencils and erasers? (2023)

A. 6
B. 12
C. 18
D. 24

Solution

The maximum number of students is the HCF of 36 and 48, which is 12.

Correct Answer: B — 12

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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)

A. 12
B. 6
C. 8
D. 10

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. A teacher has 48 pencils and 60 erasers. What is the largest number of students that can receive the same number of pencils and erasers? (2023)

A. 6
B. 12
C. 8
D. 10

Solution

The HCF of 48 and 60 is 12. Therefore, the largest number of students is 12.

Correct Answer: B — 12

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Q. If the HCF of 18 and 24 is 6, what is the LCM of these two numbers? (2023)

A. 72
B. 36
C. 48
D. 60

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)

A. 72
B. 48
C. 60
D. 84

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

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

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

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

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

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.

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

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

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

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)

A. 30
B. 25
C. 50
D. 45

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

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

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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HCF & LCM

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