Finding HCF of Products: 12x15, 15x21, 21x12
To find the Highest Common Factor (HCF) of the given products, we first express each product in terms of its prime factors.
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First Product:
$12 \times 15 = (2^2 \times 3) \times (3 \times 5) = 2^2 \times 3^2 \times 5$
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Second Product:
$15 \times 21 = (3 \times 5) \times (3 \times 7) = 3^2 \times 5 \times 7$
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Third Product:
$21 \times 12 = (3 \times 7) \times (2^2 \times 3) = 2^2 \times 3^2 \times 7$
Identifying Common Prime Factors
Now, we identify the prime factors that are common to all three prime factorizations and the lowest power of each common factor.
- The prime factor 2 appears in the first and third products ($2^2$), but not the second.
- The prime factor 3 appears in all three products as $3^2$.
- The prime factor 5 appears in the first and second products, but not the third.
- The prime factor 7 appears in the second and third products, but not the first.
The only prime factor common to all three expressions is 3, and its lowest power across all is $3^2$.
Calculating the HCF
The HCF is the product of the common prime factors raised to their lowest powers.
HCF = $3^2 = 9$
Therefore, the HCF of $12 \times 15, 15 \times 21, 21 \times 12$ is 9.
Similar Questions
- The H.C.F. and the L.C.M. of two numbers are 5 and 495, respectively. If one of the numbers is 55, find the other one.
- The HCF of two numbers is 16 and their LCM is 240. If one number is 48, find the other number.
- If the LCM of $20x^3y^2$ and $10x^4y^4$ is $20x^4y^4$, find the HCF.
- There are four table clocks. They ring every 10 min, 15 min, 20 min and 25 min respectively. If they all ring together at 10 a.m., then at what time will they ring together again?
- How many numbers less than 10000 are there which are exactly divisible by 21, 35 and 63?
- A, B and C begin together to move around a circular stadium and they complete their revolutions in 42 s, 63 s and 84 s respectively. After how much time will they come together at the starting point?
- The HCF and the LCM of two numbers are 17 and 1224, respectively. If one of the numbers is 136, find the other one.
- The LCM of two numbers is 721, and the numbers are in the ratio of 1 : 7. What is the sum of the numbers?
What is the LCM of $\sqrt[2]{169}$, $\sqrt[3]{27}$, $\sqrt[3]{64}$ and $\sqrt[2]{144}$ ?
- The HCF of $28\ p^5q^2$ and $70\ p^3q^4$ is:
Important Questions from LCM and HCF
The HCF and LCM of two numbers are 12 and 72, respectively. If the ratio of the two numbers is 2 ∶ 3, then the larger of the two numbers is:
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
Joseph visits the club on every 5 th day, Harsh visits on every 24 th day, while Sumit visits on every 9 th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
The sum of two numbers is 1215 and their HCF is 81. How many such pairs of numbers can be formed?
The LCM of two numbers in 48. Their ratio is 2:3 What is the sum of the numbers?
