Maharashtra Board Class 7 Chapter 3 Set 12 HCF and LCM Solutions

Question 1. (i) 25, 40
Answer:

\( \therefore 25 = 5 \times 5 \)
\( 40 = 2 \times 2 \times 2 \times 5 \)
\( \therefore \text{HCF of } 25 \text{ and } 40 = 5 \)
In simple words: To find the HCF, we list the prime factors of each number and identify the common factors. The product of these common prime factors gives the HCF. In this case, 5 is the only common prime factor.

🎯 Exam Tip: Always show the prime factorization steps clearly to get full marks for HCF problems.

Question 1. (ii) 56, 32
Answer:

\( \therefore 56 = 2 \times 2 \times 2 \times 7 \)
\( 32 = 2 \times 2 \times 2 \times 2 \times 2 \)
\( \therefore \text{HCF of } 56 \text{ and } 32 = 2 \times 2 \times 2 \)
\( \therefore \text{HCF of } 56 \text{ and } 32 = 8 \)
In simple words: The HCF of 56 and 32 is found by multiplying their common prime factors, which are three 2s, resulting in 8.

🎯 Exam Tip: Make sure to list all prime factors correctly for each number before identifying the common ones.

Question 1. (iii) 40, 60, 75
Answer:

\( \therefore 40 = 2 \times 2 \times 2 \times 5 \)
\( 60 = 2 \times 2 \times 3 \times 5 \)
\( 75 = 3 \times 5 \times 5 \)
\( \therefore \text{HCF of } 40, 60 \text{ and } 75 = 5 \)
In simple words: For three numbers, the HCF is the product of prime factors common to all three. Here, only the prime factor 5 is common to 40, 60, and 75.

🎯 Exam Tip: When finding HCF of three or more numbers, ensure the common factors are present in *all* the numbers.

Question 1. (iv) 16, 27
Answer:

\( \therefore 16 = 2 \times 2 \times 2 \times 2 \times 1 \)
\( 27 = 3 \times 3 \times 3 \times 1 \)
\( \therefore \text{HCF of } 16 \text{ and } 27 = 1 \)
In simple words: Since 16 and 27 have no common prime factors other than 1, their HCF is 1. Such numbers are called coprime.

🎯 Exam Tip: Remember that if two numbers have no common prime factors, their HCF is always 1.

Question 1. (v) 18, 32,48
Answer:

\( \therefore 18 = 2 \times 3 \times 3 \)
\( 32 = 2 \times 2 \times 2 \times 2 \times 2 \)
\( 48 = 2 \times 2 \times 2 \times 2 \times 3 \)
\( \therefore \text{HCF of } 18, 32 \text{ and } 48 = 2 \)
In simple words: The only prime factor common to 18, 32, and 48 is 2, making their HCF equal to 2.

🎯 Exam Tip: Systematically list prime factors for all numbers to avoid missing any common factors.

Question 1. (vi) 105, 154
Answer:

\( \therefore 105 = 3 \times 5 \times 7 \)
\( 154 = 2 \times 7 \times 11 \)
\( \therefore \text{HCF of } 105 \text{ and } 154 = 7 \)
In simple words: By finding the prime factors, we see that 7 is the only common factor between 105 and 154, so their HCF is 7.

🎯 Exam Tip: Practice identifying prime factors quickly to save time during exams.

Question 1. (vii) 42, 45, 48
Answer:

\( \therefore 42 = 2 \times 3 \times 7 \)
\( 45 = 3 \times 3 \times 5 \)
\( 48 = 2 \times 2 \times 2 \times 2 \times 3 \)
\( \therefore \text{HCF of } 42, 45 \text{ and } 48 = 3 \)
In simple words: The common prime factor for 42, 45, and 48 is 3, making it their HCF.

🎯 Exam Tip: Double-check your prime factorization for each number before finding common factors.

Question 1. (viii) 57, 75, 102
Answer:

\( \therefore 57 = 3 \times 19 \)
\( 75 = 3 \times 5 \times 5 \)
\( 102 = 2 \times 3 \times 17 \)
\( \therefore \text{HCF of } 57, 75 \text{ and } 102 = 3 \)
In simple words: By factoring each number, we find that 3 is the only prime factor common to 57, 75, and 102.

🎯 Exam Tip: Be careful with larger numbers; ensure you use prime numbers for division during factorization.

Question 1. (ix) 56, 57
Answer:

\( \therefore 56 = 2 \times 2 \times 2 \times 7 \times 1 \)
\( 57 = 3 \times 19 \times 1 \)
\( \therefore \text{HCF of } 56 \text{ and } 57 = 1 \)
In simple words: Since 56 and 57 share no common prime factors other than 1, their HCF is 1, indicating they are coprime.

🎯 Exam Tip: Always remember that the HCF of any two consecutive natural numbers is 1.

Question 1. (x) 777, 315, 588
Answer:

\( \therefore 777 = 3 \times 7 \times 37 \)
\( 315 = 3 \times 3 \times 5 \times 7 \)
\( 588 = 2 \times 2 \times 3 \times 7 \times 7 \)
\( \therefore \text{HCF of } 777, 315 \text{ and } 588 = 3 \times 7 \)
\( \text{HCF of } 777, 315 \text{ and } 588 = 21 \)
In simple words: The common prime factors for 777, 315, and 588 are 3 and 7. Multiplying them gives an HCF of 21.

🎯 Exam Tip: When dealing with larger numbers, breaking them down into prime factors step-by-step is crucial for accuracy.

Question 2.
Find the HCF by the division method and reduce to the simplest form:
(i) \( \frac{275}{525} \)
Answer:

ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक भागफल विधि है जो 275 और 525 का महत्तम समापवर्तक (HCF) ज्ञात करती है। 525 को 275 से विभाजित किया जाता है, शेषफल 250 प्राप्त होता है। फिर 275 को 250 से विभाजित किया जाता है, शेषफल 25 प्राप्त होता है। अंत में, 250 को 25 से विभाजित किया जाता है, जिससे शेषफल 0 प्राप्त होता है। अंतिम भाजक 25 ही HCF है।
\( \text{HCF of } 275 \text{ and } 525 = 25 \)
\( \frac{275}{525} = \frac{275 \div 25}{525 \div 25} = \frac{11}{21} \)
In simple words: Using the division method, we find the HCF of 275 and 525 to be 25. Dividing both the numerator and denominator by 25 reduces the fraction to its simplest form, \( \frac{11}{21} \).

🎯 Exam Tip: The division method is efficient for finding the HCF of two numbers and is often required for reducing fractions to their simplest form.

Question 2.
(ii) \( \frac{76}{133} \)
Answer:

ℹ️ चित्र व्याख्या (Diagram Explanation): यह भागफल विधि का उपयोग करके 76 और 133 का HCF निकालने की प्रक्रिया दर्शाती है। 133 को 76 से विभाजित करने पर शेषफल 57 मिलता है। फिर 76 को 57 से विभाजित करने पर शेषफल 19 मिलता है। अंत में, 57 को 19 से विभाजित करने पर शेषफल 0 मिलता है, जिससे HCF 19 प्राप्त होता है।
\( \text{HCF of } 76 \text{ and } 133 = 19 \)
\( \frac{76}{133} = \frac{76 \div 19}{133 \div 19} = \frac{4}{7} \)
In simple words: By the division method, the HCF of 76 and 133 is 19. Dividing both parts of the fraction by 19 simplifies it to \( \frac{4}{7} \).

🎯 Exam Tip: Always state the HCF explicitly before using it to simplify the fraction.

Question 2.
(iii) \( \frac{161}{69} \)
Answer:

ℹ️ चित्र व्याख्या (Diagram Explanation): यह आरेख 161 और 69 का HCF ज्ञात करने के लिए यूक्लिडियन एल्गोरिथम (भागफल विधि) का उपयोग करता है। 161 को 69 से विभाजित करने पर शेषफल 23 बचता है। फिर 69 को 23 से विभाजित करने पर शेषफल 0 प्राप्त होता है। इस प्रकार, अंतिम भाजक 23 ही HCF है।
\( \text{HCF of } 161 \text{ and } 69 = 23 \)
\( \frac{161}{69} = \frac{161 \div 23}{69 \div 23} = \frac{7}{3} \)
In simple words: The division method reveals that the HCF of 161 and 69 is 23. Dividing both the numerator and denominator by 23 simplifies the fraction to \( \frac{7}{3} \).

🎯 Exam Tip: When simplifying fractions using HCF, ensure the division is accurate for both the numerator and denominator.

Maharashtra Board Class 7 Maths Chapter 3 Hcf And Lcm Practice Set 12 Intext Questions And Activities

Question 1.
In each of the following examples, write all the factors of the numbers and find the greatest common divisor. (Textbook pg. no. 17)
(i) 28, 42
(ii) 51, 27
(iii) 25, 15, 35
Answer:
Solution:
(i) Factors of 28 = 1,2,4, 7, 14, 28
Factors of 42 = 1,2, 3, 6, 7, 14, 21, 42
\( \therefore \text{HCF of } 28 \text{ and } 42 = 14 \)
In simple words: By listing all factors, we find the common factors for 28 and 42. The largest common factor, or HCF, is 14.

🎯 Exam Tip: Listing factors systematically helps avoid missing any common divisors. Always check for the largest among the common factors.

Question 1.
(ii) 51, 27
Answer:
Factors of 51 = 1, 3, 17, 51
Factors of 27 = 1, 3, 9, 27
\( \therefore \text{HCF of } 51 \text{ and } 27 = 3 \)
In simple words: Listing the factors of 51 and 27 shows that 3 is the greatest common factor.

🎯 Exam Tip: For smaller numbers, factor listing can be a straightforward method, but for larger numbers, prime factorization or the division method is more efficient.

Question 1.
(iii) 25, 15, 35
Answer:
Factors of 25 = 1, 5, 25
Factors of 15 = 1, 3, 5, 15
Factors of 35 = 1, 5, 7, 35
\( \therefore \text{HCF of } 25, 15 \text{ and } 35 = 5 \)
In simple words: The factors common to 25, 15, and 35 are 1 and 5. The greatest among these is 5, which is the HCF.

🎯 Exam Tip: When finding the HCF for three or more numbers by listing factors, ensure the common factors are shared by ALL numbers.