Mathematics

HCF and LCM using prime factorisation

अभाज्य गुणनखंडन द्वारा महत्तम समापवर्तक और लघुत्तम समापवर्त्य

In this Class 10 Mathematics topic from Real Numbers, students learn to find the HCF and LCM of two or more numbers through prime factorisation. They break each number into prime factors, compare the resulting powers, and select the appropriate common factors: the lowest powers for HCF and the highest powers for LCM. The topic also develops accuracy in writing factor trees, simplifies comparison between methods, and helps students apply these concepts to numerical and everyday problem-solving situations.

Practice questions

01 If the prime factorisations of two numbers are (2^3\times3^2\times5) and (2^2\times3^3\times7), what is their HCF?

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Answer and explanation

HCF and LCM using prime factorisation

Practice questions

01 If the prime factorisations of two numbers are (2^3\times3^2\times5) and (2^2\times3^3\times7), what is their HCF?

02 If the HCF of two numbers is (18), their LCM is (540), and one number is (90), what is the other number?

03 The prime factorisations of three numbers are (2^4\times3\times5^2), (2^2\times3^3\times5), and (2^3\times3^2\times7). What is their LCM?

04 The product of two numbers is (12960) and their HCF is (36). What is their LCM?

05 If (a=2^5\times3^2\times11) and (b=2^3\times3^4\times5), what is the value of (\frac{\text{LCM}}{\text{HCF}})?

06 What is the smallest number exactly divisible by (24), (36), and (54)?

07 What is the greatest number that can exactly divide (168), (252), and (420)?

08 Two numbers are (2^4\times3^2) and (2^2\times3\times5^2). The product of their HCF and LCM will be equal to what?

09 If (x=2^a\times3^4\times5) and (y=2^3\times3^b\times7) have HCF (2^2\times3^2), which statement about (a) and (b) is correct?

10 If the HCF of two numbers is (12) and their LCM is (420), which pair can be such numbers?

11 A school has (72) boys and (96) girls. They must be arranged in separate rows with the same number of students in each row, and this number must be maximum. How many students will be in each row?

12 Three bells ring at intervals of (18), (24), and (30) seconds respectively. If they ring together now, after how many seconds will they ring together again?

13 If the LCM of two numbers is (2^5\times3^3\times5) and their HCF is (2^2\times3), what is the power of (2) in the product of the two numbers?

14 If two numbers are coprime and their product is (391), what is their LCM?

15 Which of the following numbers is divisible by both (2^3\times3^2\times5) and (2^2\times3^4\times7)?

16 Two numbers are (180) and (252). What is their HCF in prime factor form?

17 If (96=2^5\times3) and (144=2^4\times3^2), what is the LCM of (96) and (144)?

18 A number (N) has prime factorisation (2^4\times3^2\times5). What is the HCF of (N) and (2^3\times3^5\times7)?

19 If the HCF of two numbers is (15) and their LCM is (630), what is the product of the two numbers?

20 If (A=2^2\times3\times5^3) and (B=2^5\times3^2\times5), which prime factor will have power (3) in the LCM?

21 A shopkeeper has (126) red pens and (210) blue pens. He wants to make the maximum number of identical packets so that each packet has the same number of red pens and the same number of blue pens. How many packets can he make?

22 What is the smallest number that leaves remainder (0) when divided by (45), (60), and (75)?

23 The HCF of two numbers is (2^2\times3) and their LCM is (2^4\times3^3\times5). If one number is (2^4\times3\times5), what is the other number?

24 If two numbers have no common prime factor in their prime factorisations, which statement about their HCF is correct?

25 Which statement about (84) and (198) is correct?

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