🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books

Hcf And Lcm Quiz 3 (10 MCQs)

This set of multiple-choice questions evaluates understanding of factorization, LCM calculation, and related concepts such as greatest common factor, prime factorization, and divisibility rules. It tests the ability to find common factors, determine the highest power of factors, and apply set theory principles to solve problems involving common multiples and the least common multiple.

Quiz Instructions

Select an option to see the correct answer instantly.

1. What is the least common denominator between 2/5 and 1/2?

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

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: The least common denominator (LCD) is the least common multiple (LCM) of the denominators of the fractions. The denominators here are 5 and 2.

Common Mistakes: A common mistake is to confuse the least common denominator with the highest common factor (HCF) or to simply choose the larger denominator without finding the LCM.

Explanations: To find the least common denominator between 2/5 and 1/2, we need to determine the least common multiple (LCM) of the denominators 5 and 2. The multiples of 5 are 5, 10, 15, 20, etc., and the multiples of 2 are 2, 4, 6, 8, 10, etc. The smallest number that appears in both lists is 10. Therefore, the least common denominator is 10.

Option Analysis:

Option A: 5 is not the least common denominator because it is not a multiple of 2.
Option B: 10 is the least common denominator because it is the smallest number that is a multiple of both 5 and 2.
Option C: 6 is not the least common denominator because it is not a multiple of 5.
Option D: 2 is not the least common denominator because it is not a multiple of 5.

2. What is the Greatest Common Factor of 15 and 12?

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

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

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), which is a different concept.

Explanations: To find the GCF of 15 and 12, we list the factors of each number: - Factors of 15: 1, 3, 5, 15 - Factors of 12: 1, 2, 3, 4, 6, 12 The common factors of 15 and 12 are 1 and 3. The greatest of these common factors is 3. Therefore, the GCF of 15 and 12 is 3.

Option Analysis:

Option A: 1 is a common factor but not the greatest.
Option B: 3 is the greatest common factor.
Option C: 5 is not a factor of 12.
Option D: 50 is not a factor of either 15 or 12.

3. Suppose A=\{factors of 20\} and B=\{factors of 12\}, what is $A\cap B$

Correct Answer: Correct answer is: (A) \{1, 2, 4\}.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: The intersection of two sets, denoted by $A \cap B$, contains all elements that are common to both sets A and B.

Common Mistakes: A common mistake is to confuse the intersection with the union of sets, which includes all elements from both sets without removing duplicates.

Explanations: To find the intersection of sets A and B, we need to identify the common factors of 20 and 12. The factors of 20 are \{1, 2, 4, 5, 10, 20\} and the factors of 12 are \{1, 2, 3, 4, 6, 12\}. The common factors are \{1, 2, 4\}. Therefore, the intersection of A and B is \{1, 2, 4\}.

Option Analysis:

Option A: Correct. The common factors of 20 and 12 are \{1, 2, 4\}.
Option B: Incorrect. This set includes factors of 20 and 12 but also includes factors that are not common to both.
Option C: Incorrect. This set includes only the numbers 12 and 20, which are not common factors.
Option D: Incorrect. This set includes factors of 20 and 12 but also includes factors that are not common to both.

4. What is the least common multiple between 3 and 9?

Correct Answer: Correct answer is: (B) 9.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Easy

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 3 and 9, we list the multiples of each number: Multiples of 3: 3, 6, 9, 12, 15, 18, ... Multiples of 9: 9, 18, 27, 36, ... The smallest number that appears in both lists is 9. Therefore, the LCM of 3 and 9 is 9.

Option Analysis:

Option A: 3 is not the LCM because it is not divisible by 9.
Option B: 9 is the LCM because it is the smallest number divisible by both 3 and 9.
Option C: 12 is not the LCM because it is not divisible by 9.
Option D: 27 is not the LCM because it is not the smallest number divisible by both 3 and 9.

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

5. The HCF and LCM of the two numbers are 13 and 1989 respectively. If one of the numbers is 117, then find the sum of both the numbers

Correct Answer: Correct answer is: (C) 338.

Exam Relevance: GRE, GMAT, CAT, SAT

Difficulty: Moderate

Concept notes: The relationship between the HCF, LCM, and the two numbers is given by the formula: HCF × LCM = Product of the two numbers.

Common Mistakes: A common mistake is to forget the relationship between HCF, LCM, and the product of the two numbers, leading to incorrect calculations.

Explanations: Given: - HCF = 13 - LCM = 1989 - One of the numbers = 117 Using the formula HCF × LCM = Product of the two numbers: 13 × 1989 = 117 × Other number First, calculate the product of HCF and LCM: 13 × 1989 = 25857 Now, solve for the other number: 25857 = 117 × Other number Other number = 25857 ÷ 117 = 221 Sum of both numbers: 117 + 221 = 338

Option Analysis:

Option A: 119 is incorrect because it does not match the sum of the two numbers.
Option B: 221 is incorrect because it is only one of the numbers, not the sum.
Option C: 338 is correct as it is the sum of the two numbers.
Option D: 439 is incorrect because it does not match the sum of the two numbers.

6. Find the least number which is exactly divisible by 12, 15, and 20. *

Correct Answer: Correct answer is: (B) 60.

Exam Relevance: SAT, GRE, GMAT, CAT

Difficulty: Easy

Concept notes: The least number exactly divisible by 12, 15, and 20 is the Least Common Multiple (LCM) of these numbers.

Common Mistakes: A common mistake is to confuse the LCM with the Highest Common Factor (HCF), which is not the correct approach for finding the least number divisible by multiple numbers.

Explanations: To find the LCM of 12, 15, and 20, we first factorize each number into its prime factors: - 12 = 2^2 * 3 - 15 = 3 * 5 - 20 = 2^2 * 5 The LCM is found by taking the highest power of each prime factor present in the factorizations: - The highest power of 2 is 2^2. - The highest power of 3 is 3. - The highest power of 5 is 5. Thus, the LCM is 2^2 * 3 * 5 = 4 * 3 * 5 = 60. Therefore, the least number exactly divisible by 12, 15, and 20 is 60.

Option Analysis:

Option A: 50 is not divisible by 12, 15, and 20.
Option B: 60 is divisible by 12, 15, and 20.
Option C: 70 is not divisible by 12, 15, and 20.
Option D: 65 is not divisible by 12, 15, and 20.

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

7. Find the LCM of (a3-b3), (a2-b2), (a-b).

Correct Answer: Correct answer is: (B) (a3-b3) (a + b).

Exam Relevance: GRE, GMAT, SAT, ACT

Difficulty: Moderate

Concept notes: The LCM of algebraic expressions is found by taking the highest power of each factor present in the expressions.

Common Mistakes: A common mistake is to overlook the factorization of the expressions and directly take the highest power of each term without considering the factorized form.

Explanations: To find the LCM of (a3-b3), (a2-b2), and (a-b), we first factorize each expression: - (a3-b3) = (a-b)(a2+ab+b2) - (a2-b2) = (a-b)(a+b) - (a-b) = (a-b) The LCM is determined by taking the highest power of each factor present in the expressions: - The factor (a-b) appears in all three expressions. - The factor (a2+ab+b2) appears in (a3-b3). - The factor (a+b) appears in (a2-b2). Thus, the LCM is (a-b)(a2+ab+b2)(a+b) = (a3-b3)(a+b).

Option Analysis:

Option A: (a-b) is a factor of all expressions but not the LCM.
Option B: (a3-b3)(a+b) is the correct LCM as it includes all factors with their highest powers.
Option C: (a3-b3) is a factor but not the LCM as it does not include (a+b).
Option D: (a3-b3)(a2-b2) includes (a2-b2) which is not the highest power of (a+b).

8. What's the greatest common factor for 12 and 24?

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

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

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 12 and 24, we list the factors of each number: - Factors of 12: 1, 2, 3, 4, 6, 12 - Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 The common factors of 12 and 24 are 1, 2, 3, 4, 6, and 12. The greatest of these common factors is 12. Therefore, the GCF of 12 and 24 is 12.

Option Analysis:

Option A: 48 is not a factor of 12, so it cannot be the GCF.
Option B: 24 is a factor of 24 but not of 12, so it cannot be the GCF.
Option C: 6 is a common factor of 12 and 24, but it is not the greatest common factor.
Option D: 12 is the greatest common factor of 12 and 24.

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

9. In a morning walk, three persons step off together. Their steps measure 80cm, 85cm and 90cm respectively. What is the minimum distance each should walk so that all can cover the same distance in complete steps?

Correct Answer: Correct answer is: (B) 12240cm.

Exam Relevance: GRE, GMAT, SAT, ACT

Difficulty: Moderate

Concept notes: The problem requires finding the least common multiple (LCM) of the step lengths to determine the minimum distance each person should walk so that all can cover the same distance in complete steps.

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 minimum distance each person should walk, we need to calculate the LCM of 80cm, 85cm, and 90cm. 1. Prime factorize each number: - 80 = 2^4 × 5 - 85 = 5 × 17 - 90 = 2 × 3^2 × 5 2. Identify the highest power of each prime factor: - For 2: 2^4 - For 3: 3^2 - For 5: 5 - For 17: 17 3. Multiply these highest powers together to get the LCM: LCM = 2^4 × 3^2 × 5 × 17 = 16 × 9 × 5 × 17 = 12240 Thus, the minimum distance each person should walk is 12240cm.

Option Analysis:

Option A: 12400cm is incorrect because it is not the LCM of 80, 85, and 90.
Option B: 12240cm is the correct LCM of 80, 85, and 90.
Option C: 11240cm is incorrect because it is not the LCM of 80, 85, and 90.
Option D: 12440cm is incorrect because it is not the LCM of 80, 85, and 90.

Mnemonic: LCM: Least Common Multiple

10. The least number which when divided by 4, 6, 8, 12 and 16 leaves a remainder of 2 in each case is

Correct Answer: Correct answer is: (C) 50.

Exam Relevance: GRE, GMAT, CAT, SAT

Difficulty: Moderate

Concept notes: To find the least number that leaves a remainder of 2 when divided by 4, 6, 8, 12, and 16, we first need to find the least common multiple (LCM) of these numbers and then add the remainder.

Common Mistakes: A common mistake is to directly find the LCM of the numbers and not add the remainder of 2.

Explanations: 1. Find the LCM of 4, 6, 8, 12, and 16. - Prime factorization: - 4 = 2^2 - 6 = 2 × 3 - 8 = 2^3 - 12 = 2^2 × 3 - 16 = 2^4 - LCM is the highest power of each prime factor: 2^4 × 3 = 48. 2. Add the remainder 2 to the LCM: 48 + 2 = 50.

Option Analysis:

Option A: 46 is incorrect because it does not satisfy the condition of leaving a remainder of 2 when divided by all the given numbers.
Option B: 48 is incorrect because it is the LCM of the numbers but does not include the remainder of 2.
Option C: 50 is correct because it is the LCM of the numbers plus the remainder of 2.
Option D: 56 is incorrect because it does not satisfy the condition of leaving a remainder of 2 when divided by all the given numbers.

Frequently Asked Questions

What is the difference between HCF and LCM?

HCF (Highest Common Factor) is the largest number that divides two or more numbers without leaving a remainder, while LCM (Least Common Multiple) is the smallest number that is a multiple of two or more numbers.

How do you find the HCF of two numbers?

To find the HCF of two numbers, you can use the prime factorization method or the Euclidean algorithm, which involves dividing the larger number by the smaller one and then continuing with the remainder until the remainder is zero.

What is the significance of LCM in real-world applications?

LCM is used in various real-world applications, such as scheduling events that repeat at different intervals, finding the least common denominator for adding or subtracting fractions, and determining the least common period for repeating patterns.

Can HCF and LCM be applied to algebraic expressions?

Yes, HCF and LCM can be applied to algebraic expressions. The process involves finding the common factors or multiples of the terms in the expressions, similar to how it is done with numbers.

How does understanding HCF and LCM help in problem-solving?

Understanding HCF and LCM helps in problem-solving by providing methods to find common factors and multiples, which can simplify complex problems involving ratios, proportions, and divisibility.

🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books