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Hcf And Lcm Quiz 2 (10 MCQs)

This set of multiple-choice questions evaluates understanding of factors, common factors, greatest common factor (GCF), least common multiple (LCM), and their applications in fraction operations, time synchronization, and prime factorization. Skills tested include identifying common factors, calculating LCM, and applying divisibility rules.

Quiz Instructions

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1. Which are the factors of both 6 and 15?

Correct Answer: Correct answer is: (D) 3.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Easy

Concept notes: The factors of a number are the integers that divide the number without leaving a remainder. The highest common factor (HCF) of two numbers is the largest number that divides both of them without leaving a remainder.

Common Mistakes: A common mistake is to confuse factors with multiples or to overlook the common factors of the given numbers.

Explanations: To find the factors of both 6 and 15, we first list the factors of each number: - Factors of 6: 1, 2, 3, 6 - Factors of 15: 1, 3, 5, 15 The common factors of 6 and 15 are the numbers that appear in both lists, which are 1 and 3. Among the given options, the only common factor is 3.

Option Analysis:

Option A: 9 is not a factor of either 6 or 15.
Option B: 5 is not a factor of 6.
Option C: 6 is not a factor of 15.
Option D: 3 is a factor of both 6 and 15.

2. What's the least common MULTIPLE for 3 and 5?

Correct Answer: Correct answer is: (B) 15.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Easy

Concept notes: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

Common Mistakes: A common mistake is to confuse the LCM with the highest common factor (HCF), which is the largest number that divides both numbers without leaving a remainder.

Explanations: To find the LCM of 3 and 5, we list the multiples of each number: Multiples of 3: 3, 6, 9, 12, 15, 18, 21, ... Multiples of 5: 5, 10, 15, 20, 25, ... The smallest number that appears in both lists is 15. Therefore, the LCM of 3 and 5 is 15.

Option Analysis:

Option A: 1 is not a multiple of either 3 or 5.
Option B: 15 is the smallest number that is a multiple of both 3 and 5.
Option C: 25 is a multiple of 5 but not of 3.
Option D: 3 is a multiple of 3 but not of 5.

3. What is the least common multiple you could use to subtract $\frac{1}{3}-\frac{1}{4}$

Correct Answer: Correct answer is: (C) 12.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. In this case, we need the LCM of 3 and 4 to subtract the fractions $\frac{1}{3} - \frac{1}{4}$.

Common Mistakes: A common mistake is to use the highest common factor (HCF) instead of the least common multiple (LCM) when dealing with fractions.

Explanations: To subtract $\frac{1}{3} - \frac{1}{4}$, we need a common denominator. The least common multiple of 3 and 4 is 12. Therefore, we convert the fractions to have a common denominator of 12: \[ \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12} \] Now, we can subtract the fractions: \[ \frac{4}{12} - \frac{3}{12} = \frac{1}{12} \] Thus, the least common multiple used is 12.

Option Analysis:

Option A: 3 is not the LCM of 3 and 4.
Option B: 4 is not the LCM of 3 and 4.
Option C: 12 is the LCM of 3 and 4.
Option D: 24 is a multiple of 3 and 4, but not the least common multiple.

Mnemonic: LCM: Least Common Multiple, think "Lowest Common Multiple"

4. What's the greatest common factor for 8 and 12?

Correct Answer: Correct answer is: (D) 4.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The greatest common factor (GCF) of two numbers is the largest number that divides both of them without leaving a remainder.

Common Mistakes: A common mistake is to confuse the GCF with the least common multiple (LCM). The GCF is the largest number that divides both numbers, while the LCM is the smallest number that is a multiple of both numbers.

Explanations: To find the GCF of 8 and 12, we first list the factors of each number: - Factors of 8: 1, 2, 4, 8 - Factors of 12: 1, 2, 3, 4, 6, 12 The common factors of 8 and 12 are 1, 2, and 4. Among these, the greatest common factor is 4.

Option Analysis:

Option A: 24 is not a factor of 8 or 12, so it cannot be the GCF.
Option B: 8 is a factor of 8 but not of 12, so it cannot be the GCF.
Option C: 12 is a factor of 12 but not of 8, so it cannot be the GCF.
Option D: 4 is a factor of both 8 and 12, and it is the greatest common factor.

Mnemonic: GCF: Greatest Common Factor, think "Greatest Common Divisor" (GCD).

5. Least Common Multiple

Correct Answer: Correct answer is: (B) Used to find a common denominator so we can add or subtract fractions.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The least common multiple (LCM) is a fundamental concept in arithmetic used to find a common denominator when adding or subtracting fractions.

Common Mistakes: A common mistake is confusing the LCM with the highest common factor (HCF), which is used for simplifying fractions rather than adding or subtracting them.

Explanations: The least common multiple (LCM) is used to find a common denominator so we can add or subtract fractions. When adding or subtracting fractions with different denominators, we need a common denominator to perform the operation. The LCM of the denominators provides the smallest common denominator that can be used for this purpose. This ensures that the fractions can be expressed with the same denominator, allowing for straightforward addition or subtraction.

Option Analysis:

Option A: The LCM is not used to find the answer to adding directly; it is used to find a common denominator for the fractions being added.
Option B: The LCM is used to find a common denominator so we can add or subtract fractions, which is the correct application.
Option C: The LCM is not used to find a common numerator; it is used to find a common denominator.
Option D: The LCM does not make the top and bottom the same; it makes the denominators the same for the purpose of adding or subtracting fractions.

Mnemonic: LCM: Least Common Multiple, LCM: Lowest Common Denominator

6. What is the greatest common factor of 72 and 84?

Correct Answer: Correct answer is: (E) 12.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The greatest common factor (GCF) of two numbers is the largest number that divides both of them without leaving a remainder. To find the GCF, we can use the prime factorization method.

Explanations: To find the GCF of 72 and 84, we first find their prime factorizations: - Prime factorization of 72: $72 = 2^3 \times 3^2$ - Prime factorization of 84: $84 = 2^2 \times 3 \times 7$ Next, we identify the common prime factors and take the lowest power of each: - Common prime factors: 2 and 3 - Lowest power of 2: $2^2$ - Lowest power of 3: $3^1$ Thus, the GCF is $2^2 \times 3 = 4 \times 3 = 12$.

Option Analysis:

Option A: 2 is a common factor but not the greatest.
Option B: 3 is a common factor but not the greatest.
Option C: 4 is a common factor but not the greatest.
Option D: 6 is a common factor but not the greatest.
Option E: 12 is the greatest common factor.

Mnemonic: GCF: Greatest Common Factor, "Greatest" means the largest number that divides both.

7. What's the least common Multiple for 6 and 4?

Correct Answer: Correct answer is: (C) 12.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Moderate

Concept notes: The least common multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers.

Common Mistakes: A common mistake is to confuse the LCM with the highest common factor (HCF), which is the largest number that divides both numbers without leaving a remainder.

Explanations: To find the LCM of 6 and 4, we can use the prime factorization method: - Prime factorization of 6: $2 \times 3$ - Prime factorization of 4: $2^2$ The LCM is found by taking the highest power of each prime that appears in the factorizations: - The highest power of 2 is $2^2$ - The highest power of 3 is $3^1$ Thus, the LCM is $2^2 \times 3 = 4 \times 3 = 12$. Therefore, the least common multiple of 6 and 4 is 12.

Option Analysis:

Option A: 6 is not the LCM of 6 and 4 because 4 does not divide 6 evenly.
Option B: 48 is not the LCM of 6 and 4 because it is larger than necessary; 12 is the smallest number divisible by both 6 and 4.
Option C: 12 is the correct LCM of 6 and 4 because it is the smallest number divisible by both 6 and 4.
Option D: 8 is not the LCM of 6 and 4 because 6 does not divide 8 evenly.

Mnemonic: LCM: "Least Common Multiple" - "Lowest Common Multiple"

8. Three bicyclists ride laps around a trail. Liam completes 1 lap every 4 minutes, Mason completes 1 lap every 5 minutes, and William completes 1 lap every 6 minutes. How many laps does Liam complete before all three bicyclists are at the beginning of the trail at the same time?

Correct Answer: Correct answer is: (C) 15

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The problem involves finding the least common multiple (LCM) of the times taken by each bicyclist to complete one lap. The LCM will give the time when all three bicyclists will be at the beginning of the trail at the same time. We then use this time to determine how many laps Liam completes.

Common Mistakes: A common mistake is to confuse the LCM with the highest common factor (HCF) or to incorrectly calculate the LCM.

Explanations: To find the time when all three bicyclists are at the beginning of the trail at the same time, we need to determine the least common multiple (LCM) of their lap times. The lap times are 4 minutes, 5 minutes, and 6 minutes. First, we find the prime factorizations: - 4 = 2^2 - 5 = 5^1 - 6 = 2^1 * 3^1 The LCM is found by taking the highest power of each prime that appears in the factorizations: - LCM = 2^2 * 5^1 * 3^1 = 4 * 5 * 3 = 60 minutes This means that all three bicyclists will be at the beginning of the trail at the same time after 60 minutes. To find out how many laps Liam completes in 60 minutes, we divide 60 by Liam's lap time: - Number of laps Liam completes = 60 / 4 = 15 laps Thus, Liam completes 15 laps before all three bicyclists are at the beginning of the trail at the same time.

Option Analysis:

Option A: 10 laps is incorrect because the LCM of 4, 5, and 6 is 60, and Liam completes 15 laps in 60 minutes.
Option B: 12 laps is incorrect because the LCM of 4, 5, and 6 is 60, and Liam completes 15 laps in 60 minutes.
Option C: 15 laps is correct because the LCM of 4, 5, and 6 is 60, and Liam completes 15 laps in 60 minutes.
Option D: 60 laps is incorrect because the LCM of 4, 5, and 6 is 60, and Liam completes 15 laps in 60 minutes.

Mnemonic: LCM: Least Common Multiple

9. What is the Least Common Multiple of 5 and 2?

Correct Answer: Correct answer is: (C) 10.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Easy

Concept notes: The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

Common Mistakes: A common mistake is to confuse the LCM with the product of the two numbers, which is not always the smallest common multiple.

Explanations: To find the LCM of 5 and 2, we list the multiples of each number: Multiples of 5: 5, 10, 15, 20, ... Multiples of 2: 2, 4, 6, 8, 10, 12, ... The smallest number that appears in both lists is 10. Therefore, the LCM of 5 and 2 is 10.

Option Analysis:

Option A: 6 is not a multiple of 5.
Option B: 20 is a multiple of both 5 and 2, but it is not the smallest.
Option C: 10 is the smallest number that is a multiple of both 5 and 2.
Option D: 25 is not a multiple of 2.

10. The ratio of two numbers is 3: 4 and their H.C.F. is 4. Their L.C.M. is:

Correct Answer: Correct answer is: (D) 48.

Exam Relevance: GMAT, GRE, SAT, CAT

Difficulty: Moderate

Concept notes: The H.C.F. (Highest Common Factor) of two numbers is the largest number that divides both of them without leaving a remainder. The L.C.M. (Least Common Multiple) is the smallest number that is a multiple of both numbers. The relationship between the H.C.F. and L.C.M. of two numbers is given by the formula: H.C.F. × L.C.M. = Product of the two numbers.

Common Mistakes: A common mistake is to assume that the L.C.M. can be directly calculated from the ratio without considering the H.C.F. or the actual values of the numbers.

Explanations: Given the ratio of the two numbers is 3:4 and their H.C.F. is 4, we can express the numbers as 3x and 4x, where x is the H.C.F. Therefore, the numbers are 3 × 4 = 12 and 4 × 4 = 16. The product of the two numbers is 12 × 16 = 192. Using the relationship H.C.F. × L.C.M. = Product of the two numbers, we get 4 × L.C.M. = 192. Solving for L.C.M., we find L.C.M. = 192 / 4 = 48.

Option Analysis:

Option A: 12 is incorrect because it does not satisfy the relationship H.C.F. × L.C.M. = Product of the two numbers.
Option B: 16 is incorrect because it does not satisfy the relationship H.C.F. × L.C.M. = Product of the two numbers.
Option C: 24 is incorrect because it does not satisfy the relationship H.C.F. × L.C.M. = Product of the two numbers.
Option D: 48 is correct because it satisfies the relationship H.C.F. × L.C.M. = Product of the two numbers.

Frequently Asked Questions

What is the difference between H.C.F. and L.C.M.?

H.C.F., or Highest Common Factor, is the largest number that divides two or more numbers without leaving a remainder. L.C.M., or Least Common Multiple, is the smallest number that is a multiple of two or more numbers.

How can I find the L.C.M. of two numbers?

To find the L.C.M. of two numbers, you can list the multiples of each number until you find the smallest common multiple, or use prime factorization to identify the highest powers of all prime factors involved.

What is the practical application of H.C.F. and L.C.M.?

H.C.F. is useful in simplifying fractions and finding the greatest common divisor, while L.C.M. is used in adding or subtracting fractions with different denominators and determining the least common period for repeating events.

How do I use prime factorization to find the H.C.F.?

To find the H.C.F. using prime factorization, you first express each number as a product of its prime factors, then identify the common factors and multiply them together to get the H.C.F.

What is the relationship between H.C.F. and L.C.M. of two numbers?

The product of the H.C.F. and L.C.M. of two numbers is equal to the product of the two numbers themselves. This relationship can be used to find one if the other is known.

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