LCM and HCF

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Questions and Answers

Two gears in a machine have 36 teeth and 24 teeth, respectively. If a mark is placed on one tooth of each gear, after how many rotations of the smaller gear will the marks line up again?

The marks will line up after 3 rotations of the smaller gear. Find the LCM of 36 and 24, which is 72. Divide the LCM by the number of teeth on the smaller gear: 72 / 24 = 3.

A rectangular courtyard 18 meters long and 12 meters wide is to be paved with square tiles of the same size. What is the largest possible size (in meters) of each tile so that the courtyard is completely covered?

The largest possible size of each tile is 6 meters. Find the HCF of 18 and 12, which is 6.

Three runners are running around a circular track. They complete one lap in 120 seconds, 150 seconds, and 200 seconds, respectively. If they start at the same time and place, after how many seconds will they meet again at the starting point?

They will meet again at the starting point after 600 seconds. Find the LCM of 120, 150, and 200, which is 600.

A merchant has three bundles of ropes measuring 35 meters, 49 meters, and 63 meters. He wants to cut them into pieces of equal length. What is the greatest possible length of each piece?

<p>The greatest possible length of each piece is 7 meters. Find the HCF of 35, 49, and 63, which is 7.</p> Signup and view all the answers

Two neon signs flash at intervals of 4 seconds and 6 seconds, respectively. If they flash together at a certain moment, after how many seconds will they flash together again?

<p>The signs will flash together again after 12 seconds. Find the LCM of 4 and 6, which is 12.</p> Signup and view all the answers

What is the least number of square tiles required to pave a room 15 meters 17 cm long and 9 meters 2 cm broad?

<p>Convert the measures to centimeters, so 1517 cm and 902 cm. Find the HCF(1517,902) = 41. Area of each tile = $41 \times 41 = 1681$. Area of room = $1517 \times 902 = 1368334$. Number of tiles required = $1368334/1681 = 814$. Least number of tiles is 814.</p> Signup and view all the answers

Alice wants to distribute 84 chocolates and 90 candies equally among her friends. What is the maximum number of friends she can distribute to?

<p>The maximum number of friends Alice can distribute to is 6. Find the HCF of 84 and 90, which is 6.</p> Signup and view all the answers

Two numbers are in the ratio 3:4. If their HCF is 15, then what is the LCM of these numbers?

<p>The numbers are 45 and 60. The LCM of 45 and 60 is 180.</p> Signup and view all the answers

Two gears in a machine have 36 teeth and 48 teeth, respectively. If a mark is placed on one tooth of each gear, after how many rotations of the smaller gear will the marks align again?

<p>The marks will align again after 4 rotations of the smaller gear. We need to find the LCM of $36$ and $48$ which is $144$. Then $144/36 = 4$.</p> Signup and view all the answers

Sarah wants to plant roses and tulips in her garden. She has 60 roses and 84 tulips. If she wants to plant them in rows such that each row contains the same number of plants and only one type of plant, what is the maximum number of plants she can put in each row?

<p>The maximum number of plants in each row is 12. We must find the HCF of $60$ and $84$, which is $12$.</p> Signup and view all the answers

A warehouse has two sections. In the first section, there are 180 boxes, and in the second section, there are 210 boxes. The manager wants to stack the boxes in such a way that each stack has the same number of boxes, and the boxes are from the same section. What is the greatest number of boxes that can be placed in each stack?

<p>The greatest number of boxes that can be placed in each stack is 30. We have to calculate the HCF of $180$ and $210$, that is $30$.</p> Signup and view all the answers

Two bells ring at intervals of 24 minutes and 36 minutes, respectively. If they ring together at 9:00 AM, at what time will they ring together again?

<p>The bells will ring together again at 10:12 AM. The LCM of $24$ and $36$ is $72$ minutes. So they will ring together again in $72$ minutes.</p> Signup and view all the answers

John has two lengths of rope, one is 72 cm long and the other is 90 cm long. He wants to cut them into pieces of equal length such that no rope is wasted. What is the maximum possible length of each piece?

<p>The maximum possible length of each piece is 18 cm. This requires finding the HCF of $72$ and $90$, which is $18$.</p> Signup and view all the answers

A florist has 48 roses, 72 lilies, and 96 tulips. She wants to make bouquets with an equal number of each type of flower in each bouquet. What is the maximum number of bouquets she can make?

<p>The maximum number of bouquets she can make is 24. We must calculate the HCF of $48$, $72$, and $96$, which is $24$.</p> Signup and view all the answers

Two neon signs light up at intervals of 16 seconds and 20 seconds, respectively. If they light up together now, after how much time will they next light up together?

<p>The neon signs will light up together again after 80 seconds. Calculate the LCM of $16$ and $20$, which is $80$.</p> Signup and view all the answers

A rectangular courtyard is 20 meters long and 16 meters wide. It needs to be paved with square tiles of the same size. What is the largest size of the tile that can be used so that the courtyard is paved completely?

<p>The largest size of the tile that can be used is 4 meters. Finding the HCF of $20$ and $16$, which is $4$, provides the answer.</p> Signup and view all the answers

Flashcards

What is LCM?

The smallest number that is a multiple of two or more numbers.

What is HCF (or GCD)?

The largest number that divides two or more numbers without a remainder.

Prime Factorization Method for LCM

Express each number as a product of its prime numbers, then multiply the highest power of each prime.

Prime Factorization Method for HCF

Express each number as a product of its prime numbers, then multiply the lowest power of each prime.

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Division Method for LCM

Divide numbers by common prime factors until none exist; LCM is the product of all divisors and remaining factors.

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Division Method for HCF

Divide numbers by common prime factors until none exist; HCF is the product of all common divisors.

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LCM and HCF Relationship

For two numbers a and b, the product of their LCM and HCF is equal to the product of a and b.

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LCM Applications

Used to determine when events with different intervals will occur together again.

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What is the HCF?

The largest number that divides two or more numbers without leaving a remainder.

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What is the LCM?

The smallest number that is a multiple of two or more numbers.

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LCM Problem Keywords

Keywords like 'together again', 'simultaneously', and 'least'.

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HCF Problem Keywords

Keywords like 'maximum', 'greatest', and 'divides completely'.

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Accurate Prime Factorization

Ensuring the prime factors are correct.

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Simplifying Fractions with HCF

Divide both the numerator and denominator by their HCF.

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Adding/Subtracting Fractions with LCM

Find the least common denominator using the LCM.

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Study Notes

  • LCM (Least Common Multiple) and HCF (Highest Common Factor), also known as GCD (Greatest Common Divisor), are fundamental concepts in number theory with numerous applications in solving real-world problems.
  • LCM is the smallest number that is a multiple of two or more numbers.
  • HCF is the largest number that divides two or more numbers without leaving a remainder.

Methods to Find LCM and HCF

  • Prime Factorization Method: Express each number as a product of its prime factors.
  • LCM: Take the highest power of each prime factor that appears in any of the numbers and multiply them together.
  • HCF: Take the lowest power of each prime factor that appears in all of the numbers and multiply them together.
  • Division Method: Divide the numbers simultaneously by a common prime factor until no common factor exists.
  • LCM: Multiply all the divisors and the remaining factors.
  • HCF: Multiply all the common divisors.

LCM and HCF Relationship

  • For any two positive integers a and b: LCM(a, b) * HCF(a, b) = a * b.
  • This relationship is useful for finding the LCM if the HCF is known, or vice versa.

Real-World Problems Involving LCM and HCF

  • Problems related to traffic lights: If traffic lights at different junctions change at different intervals, the LCM can be used to determine when they will change simultaneously again.
  • Problems related to periodic events: If events occur at different intervals, the LCM can be used to determine when they will occur together again.
  • Problems related to arranging objects: HCF can be used to find the maximum number of identical groups that can be formed from different quantities of objects.
  • Problems related to tiling or paving: HCF can be used to find the largest size of square tiles that can be used to cover a rectangular area completely.

Example Problems and Solutions

  • Traffic Lights: Three traffic lights change after 36 seconds, 42 seconds, and 72 seconds respectively. If they change simultaneously at 9:00 am, at what time will they change simultaneously again?
  • Solution: Find the LCM of 36, 42, and 72. The LCM is 504 seconds, which is 8 minutes and 24 seconds. They will change simultaneously again at 9:08:24 am.
  • Bells Tolling: Four bells toll at intervals of 6, 8, 12, and 20 minutes, respectively. If they start tolling together, after how many minutes will they toll together again?
  • Solution: Find the LCM of 6, 8, 12, and 20. The LCM is 120 minutes. They will toll together again after 120 minutes.
  • Arranging Books: A librarian has 144 English books, 216 Science books, and 192 Maths books. He wants to arrange them in shelves such that each shelf has the same number of books of each subject. What is the maximum number of books each shelf can contain?
  • Solution: Find the HCF of 144, 216, and 192. The HCF is 24. Each shelf can contain a maximum of 24 books.
  • Tiling a Floor: What is the largest size of square tiles that can be used to pave a rectangular floor of size 280 cm by 168 cm completely?
  • Solution: Find the HCF of 280 and 168. The HCF is 56. The largest size of the square tile is 56 cm.

Strategies for Solving LCM and HCF Word Problems

  • Identify the Problem Type: Determine whether the problem requires finding the LCM or HCF.
  • Keywords: Look for keywords such as "together again," "simultaneously," or "least" for LCM problems, and "maximum," "greatest," or "divides completely" for HCF problems.
  • List the Given Information: Write down all the relevant numbers and units.
  • Choose the Appropriate Method: Use prime factorization or division method to find the LCM or HCF.
  • Apply the Relationship: If applicable, use the relationship LCM(a, b) * HCF(a, b) = a * b to simplify the calculation.
  • Check the Answer: Ensure the answer makes sense in the context of the problem.

Common Mistakes to Avoid

  • Confusing LCM and HCF: Understand the difference between LCM (smallest multiple) and HCF (largest factor).
  • Incorrect Prime Factorization: Ensure the prime factorization is accurate.
  • Misinterpreting the Question: Understand exactly what the question is asking before attempting to solve it.
  • Calculation Errors: Double-check all calculations to avoid mistakes.

Advanced Applications

  • Simplifying Fractions: HCF can be used to simplify fractions by dividing both the numerator and denominator by their HCF.
  • Adding and Subtracting Fractions: LCM is used to find the least common denominator when adding or subtracting fractions.
  • Problems Involving Rates and Ratios: LCM and HCF can be used to solve problems involving rates and ratios, such as determining when two runners will meet again on a circular track.

Practice Questions

  • Three friends run around a circular track and complete one round in 12, 15, and 20 minutes respectively. If they start at the same point and at the same time, after how many minutes will they meet again at the starting point?
  • A merchant has three bundles of cotton measuring 340 kg, 510 kg, and 680 kg. What is the largest weight that can be used to weigh the cotton exactly?
  • What is the least number which when divided by 6, 15, and 18 leaves a remainder of 5 in each case?

Tips for Exam Preparation

  • Understand the Concepts: Have a clear understanding of LCM and HCF.
  • Practice Regularly: Solve a variety of problems to improve problem-solving skills.
  • Memorize Formulas: Remember the relationship between LCM and HCF.
  • Manage Time Effectively: Allocate sufficient time to each problem during exams.
  • Review Mistakes: Learn from mistakes and avoid repeating them.

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