# Properties Of HCF And LCM MCQs With Explanation-Sainik School Class 6 Math Study Material Notes free pdf download

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**Properties of HCF and LCM:** For a better understanding of the concepts of LCM (Lowest Common Multiple) and HCF (Highest Common Factor), we need to recollect the terms multiples and factors. Let’s learn about LCM, HCF and the relation between HCF and LCM of natural numbers.

## Properties Of HCF And LCM MCQs |

# Properties of HCF and LCM

### Property 1

The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.

LCM × HCF = Product of two Numbers

Suppose A and B are two numbers, then.

LCM (A & B) × HCF (A & B) = A × B

**Example:** If 3 and 8 are two numbers.

LCM (3,8) = 24

HCF (3,8) = 1

LCM (3,8) x HCF (3,8) = 24 x 1 = 24

Also, 3 x 8 = 24

Hence, proved.

**Note:** This property is applicable for only two numbers.

## Example 1: Show that the LCM (6, 15) × HCF (6, 15) = Product(6, 15)

**Solution:** LCM and HCF of 6 and 15:

6 = 2 × 3

15 = 3 x 5

LCM of 6 and 15 = 30

HCF of 6 and 15 = 3

LCM (6, 15) × HCF (6, 15) = 30 × 3 = 90

Product of 6 and 15 = 6 × 15 = 90

Hence, LCM (6, 15) × HCF (6, 15)=Product(6, 15) = 90

### Property 2

HCF of co-prime numbers is 1. Therefore, the LCM of given co-prime numbers is equal to the product of the numbers.

LCM of Co-prime Numbers = Product Of The Numbers

**Example:** Let us take two coprime numbers, such as 21 and 22.

LCM of 21 and 22 = 462

Product of 21 and 22 = 462

LCM (21, 22) = 21 x 22

## Example 2: 17 and 23 are two co-prime numbers. By using the given numbers verify that,

## LCM of given co-prime Numbers = Product of the given Numbers

**Solution:** LCM and HCF of 17 and 23:

17 = 1 x 7

23 = 1 x 23

LCM of 17 and 23 = 391

HCF of 17 and 23 = 1

Product of 17 and 23 = 17 × 23 = 391

Hence, LCM of co-prime numbers = Product of the numbers

### Property 3

H.C.F. and L.C.M. of Fractions:

**LCM of fractions = LCM of Numerators / HCF of Denominators**

**HCF of fractions = HCF of Numerators / LCM of Denominators**

**Example:** Let us take two fractions 4/9 and 6/21.

4 and 6 are the numerators & 9 and 12 are the denominators

LCM (4, 6) = 12

HCF (4, 6) = 2

LCM (9, 21) = 63

HCF (9, 21) = 3

Now as per the formula, we can write:

LCM (4/9, 6/21) = 12/3 = 4

HCF (4/9, 6/21) = 2/63

## Example 3: Find the LCM of the fractions 1 / 2 , 3 / 8, 3 / 4

**Solution:**

LCM of fractions = LCM of Numerators/HCF of Denominators

LCM of fractions = LCM (1,3,3)/HCF(2,8,4)=3/2

Example 4: Find the HCF of the fractions 3 / 5, 6 / 11, 9 / 20

HCF of fractions HCF of Numerators/LCM of Denominators

HCF of fractions = HCF (3,6,9)/LCM (5,11,20)=3/220

### Property 4

HCF of any two or more numbers is never greater than any of the given numbers.

Example: HCF of 4 and 8 is 4

Here, one number is 4 itself and another number 8 is greater than HCF (4, 8), i.e.,4.

### Property 5

LCM of any two or more numbers is never smaller than any of the given numbers.

Example: LCM of 4 and 8 is 8 which is not smaller to any of them.

## Solved Problems

**Example 1**: **Prove that: LCM (9 & 12) × HCF (9 & 12) = Product of 9 and 12**

**Solution:**

9 = 3 × 3 = 3²

12 = 2 × 2 × 3 = 2² × 3

LCM of 9 and 12 = 2² × 3² = 4 × 9 = 36

HCF of 9 and 12 = 3

LCM (9 & 12) × HCF (9 & 12) = 36 × 3 = 108

Product of 9 and 12 = 9 × 12 = 108

Hence, LCM (9 & 12) × HCF (9 & 12) = 9 × 12 = 108. Proved.

**Example 2:** **8 and 9 are two co-prime numbers. Using these numbers verify, LCM of Co-prime Numbers = Product Of The Numbers.**

**Solution:** LCM and HCF of 8 and 9:

8 = 2 × 2 × 2 = 2³

9 = 3 × 3 = 3²

LCM of 8 and 9 = 2³ × 3² = 8 × 9 = 72

HCF of 8 and 9 = 1

Product of 8 and 9 = 8 × 9 = 72

Hence, LCM of co-prime numbers = Product of the numbers. Therefore, verified.

**Example 3: Find the HCF of 12/25, 9/10, 18/35, 21/40.**

**Solution:**

12 = 2 × 2 × 3

9 = 3 × 3

18 = 2 × 3 × 3

21 = 3 × 7

HCF (12, 9, 18, 21) = 3

25 = 5 × 5

10 = 2 × 5

35 = 5 × 7

40 = 2 × 2 × 2 × 5

LCM(25, 10, 35, 40) = 5 × 5 × 2 × 2 × 2 × 7 = 1400

The required HCF = HCF(12, 9, 18, 21)/LCM(25, 10, 35, 40) = 3/1400

**Example 4: **If LCM and HCF of two numbers are 3 and 2 respectively, and one of the numbers is 6 then another number is?

**Solution:**

We know that LCM × HCF = a × b where a, and b are two numbers

I.e., 3 × 2 = 6 × b

Therefore, b = 1.

**Question 1:** Which of the following is NOT a property of the HCF?

## (A) The HCF of two numbers is always equal to 1.

## (B) The HCF of two numbers is always a factor of both numbers.

## (C) The product of the HCF and LCM of two numbers is always equal to the product of the two numbers.

## (D) The sum of the HCF and LCM of two numbers is always equal to the sum of the two numbers.

**Answer:** (A)

**Explanation:** The HCF of two numbers is not always equal to 1. For example, the HCF of 2 and 3 is 1, but the HCF of 6 and 12 is 6.

**Question 2:** Which of the following statements is true regarding the HCF and LCM of two numbers, a and b?

## A) HCF(a, b) is always greater than or equal to LCM(a, b).

## B) HCF(a, b) is always less than LCM(a, b).

## C) HCF(a, b) is equal to LCM(a, b).

## D) HCF(a, b) and LCM(a, b) have no specific relationship.

**Explanation:** B)

HCF(a, b) is always less than LCM(a, b). HCF is the largest common factor of two numbers, and LCM is the smallest common multiple. Since factors are smaller than multiples, the HCF is always less than the LCM.

**Question 3:** If the HCF of two numbers is 1, then what is their LCM?

## (A) 1

## (B) The product of the two numbers

## (C) The sum of the two numbers

## (D) The larger of the two numbers

**Answer:** (B)

**Explanation:** If the HCF of two numbers is 1, then the two numbers are relatively prime (have no common factors other than 1). The LCM of two relatively prime numbers is equal to the product of the two numbers.

**Question 4:** If the LCM of two numbers is equal to their product, then what is their HCF?

## (A) 1

## (B) The product of the two numbers

## (C) The sum of the two numbers

## (D) The larger of the two numbers

**Answer:** (A)

**Explanation:** If the LCM of two numbers is equal to their product, then the two numbers must be equal. The HCF of two equal numbers is equal to 1.

**Question 5:** If the HCF of two numbers is x and their LCM is y, then what is the product of the two numbers?

## (A) xy (B) x + y (C) x – y (D) xy/2

**Answer:** (A)

**Explanation:** The product of two numbers is equal to their HCF times their LCM. Therefore, the product of the two numbers is xy.

**Question 6:** If the HCF of two numbers is 1 and their sum is 10, then what are the two numbers?

## (A) 5 and 5 (B) 4 and 6 (C) 3 and 7 (D) 2 and 8

**Answer:** (C)

**Explanation:** If the HCF of two numbers is 1, then the two numbers are relatively prime (have no common factors other than 1). The only possible pair of relatively prime numbers that sum to 10 is 3 and 7.

**Question 7:** If the LCM of two numbers is 12 and their difference is 2, then what are the two numbers?

## (A) 4 and 6 (B) 3 and 7 (C) 2 and 8 (D) 1 and 9

**Answer:** (A)

**Explanation:** The LCM of two numbers is the smallest number that is a multiple of both numbers. The only possible pair of numbers that have an LCM of 12 and a difference of 2 is 4 and 6.

**Question 8:** The HCF of two numbers is 2 and their LCM is 10. What are the two numbers?

## (A) 2 and 10 (B) 2 and 5 (C) 4 and 5 (D) 6 and 10

**Answer:** (A)

**Explanation:** The product of two numbers is equal to their HCF times their LCM. Therefore, the product of the two numbers is 2 * 10 = 20. The only possible pair of numbers that have a product of 20 and an HCF of 2 is 2 and 10.

**Question 9:** The LCM of two numbers is 120 and their HCF is 2. What is the product of the two numbers?

## (A) 240 (B) 360 (C) 480 (D) 720

**Answer:** (A)

**Explanation:** The product of two numbers is equal to their HCF times their LCM. Therefore, the product of the three numbers is 2 * 120 = 240.

**Question 10:** The HCF of three numbers is 3 and their LCM is 120. What is the greatest possible product of the three numbers?

## (A) 240 (B) 360 (C) 480 (D) 720

**Answer:** (D)

**Explanation:** The product of three numbers is equal to their HCF times their LCM. Therefore, the greatest possible product of the three numbers is 3 * 120 = 360.

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**Sainik School Official Website**: Visit the official website of the Sainik School you are applying to. They often provide information about the exam pattern, syllabus, and sample question papers.**NCERT Textbooks**: The National Council of Educational Research and Training (NCERT) textbooks are widely used in Indian schools and are a valuable resource for exam preparation. Ensure you have the NCERT Math textbook for Class 6.**Solved Sample Papers and Previous Year Question Papers**: You can find solved sample papers and previous year question papers for Sainik School entrance exams at bookstores or online at ANAND CLASSES website. These papers can give you an idea of the exam pattern and types of questions asked.**Math Study Guides**: The Math study guides and reference books are publish under publication department of ANAND CLASSES and are designed to help students prepare for entrance exams. Look for books specifically tailored to the Sainik School entrance exam.**Online Resources**:**www.anandclasses.co.in**

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