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
The HCF of two numbers is (42) and their LCM is (2772). How many unordered pairs of such numbers are possible?ExpertLevel 10A number leaves remainders (43), (67), and (115) when divided by (48), (72), and (120) respectively. What is the smallest such number?ExpertLevel 10If (N) is the smallest number divisible by both (2^5\times3^2\times7) and (2^3\times3^4\times5), and (M) is the HCF of these two numbers, what is (\frac{N}{M})?ExpertLevel 10The prime factorisations of two numbers are (2^5\times3^3\times5^2) and (2^3\times3^5\times5\times11). What will be their HCF?ExpertLevel 11The prime factorisations of three numbers are (2^4\times3^2\times7), (2^2\times3^4\times5), and (2^5\times3\times11). What will be their LCM?ExpertLevel 11The HCF of two numbers is (54), their LCM is (2970), and one number is (270). What is the other number?ExpertLevel 11If (a=2^6\times3^2\times5\times13) and (b=2^3\times3^5\times5^2), what is (\frac{\text{LCM}}{\text{HCF}})?ExpertLevel 11The HCF of two numbers is (48) and their LCM is (2112). How many unordered pairs of such numbers are possible?ExpertLevel 11A number leaves remainders (29), (47), and (83) when divided by (36), (54), and (90) respectively. What is the smallest such number?ExpertLevel 11What is the greatest number that leaves the same remainder when dividing (742), (1018), and (1450)?ExpertLevel 11If (x=2^a\times3^4\times7) and (y=2^7\times3^b\times5) have HCF (2^5\times3^2), which values are possible?ExpertLevel 11If (m=2^2\times3^a\times5^3) and (n=2^6\times3^3\times5^b\times7) have LCM (2^6\times3^5\times5^4\times7), which values are correct?ExpertLevel 11A training camp has (276) students and (414) practice booklets. The maximum number of identical groups is to be formed so that each group has the same number of both separately. How many groups can be formed?ExpertLevel 11Four automatic signals ring at intervals of (18), (27), (45), and (60) seconds respectively. They ring together now. After how many seconds will they ring together again?ExpertLevel 11If the HCF of two numbers is (2^4\times3) and their LCM is (2^7\times3^4\times5^2), what will be the total power of (2) in their product?ExpertLevel 11If the HCF of (264) and (396) is (132), what is the ratio of their LCM to their HCF?ExpertLevel 11If (128), (192), and (320) are to be divided into the maximum number of equal parts, what will be the number of parts?ExpertLevel 11What is the smallest number exactly divisible by (81), (96), and (125)?ExpertLevel 11If (p=2^5\times3^2\times7) and (q=2^5\times3^2\times7\times17), which statement about (p) and (q) is correct?ExpertLevel 11The HCF of two numbers is (72) and their LCM is (4680). If the numbers are taken as (72r) and (72s), what is the value of (rs)?ExpertLevel 11