10.8
Convert Numbers to Fractions
First, represent the given decimal numbers 3.6, 0.009, and 0.27 as fractions. To do this, place the number without the decimal point over the appropriate power of 10.
- \(3.6 = \frac{36}{10}\)
- \(0.009 = \frac{9}{1000}\)
- \(0.27 = \frac{27}{100}\)
Find LCM of Numerators
Next, calculate the Least Common Multiple (LCM) of the numerators of these fractions: 36, 9, and 27.
Use prime factorization for each numerator:
- \(36 = 2^2 \times 3^2\)
- \(9 = 3^2\)
- \(27 = 3^3\)
The LCM is found by taking the highest power of each prime factor present in any of the numbers: \(LCM(36, 9, 27) = 2^2 \times 3^3 = 4 \times 27 = 108\).
Find GCD of Denominators
Then, calculate the Greatest Common Divisor (GCD) of the denominators: 10, 1000, and 100.
Use prime factorization for each denominator:
- \(10 = 2^1 \times 5^1\)
- \(1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3\)
- \(100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2\)
The GCD is found by taking the lowest power of each prime factor common to all the numbers: \(GCD(10, 1000, 100) = 2^1 \times 5^1 = 10\).
Compute LCM of Decimal Numbers
The formula for the LCM of fractions \(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\) is \(\frac{LCM(a, c, e)}{GCD(b, d, f)}\). Apply this formula to find the LCM of the original decimal numbers.
\(LCM(3.6, 0.009, 0.27) = \frac{LCM(\text{Numerators})}{\text{GCD}(\text{Denominators})} = \frac{108}{10}\)
\(LCM = 10.8\).
Similar Questions
- If HCF of 768 and \(x^6y^2\) is \(32xy\) for natural numbers \(x \ge 2, y \ge 2\), then what is the value of \((x + y)\)?
- What is the HCF of \(2^{36}-1\) and \(2^{45}-1\)?
The HCF of \(x\) and \(y\) is \(H\). Consider the following statements in respect of the HCF of \(p=\frac{x^3 + y^3}{x^2-xy+y^2}\) and \(q=\frac{x^3-y^3}{x^2+xy+y^2}\) :
I. The HCF of \(p\) and \(q\) can be \(H\).
II. The HCF of \(p\) and \(q\) can be \(2H\).
Which of the statements given above is/are correct?
- What is the HCF of \(x^3 + y^3 + 3xy-1\) and \((x + y)^2 - 1\)?
- A number N is such that when divided by 4, 6, 7 or 9, it leaves 3 as remainder. What is the smallest 4-digit number that satisfies this property?
- A plank of wood 4·25 m long and 3·4 m wide is to be cut into square pieces of equal size. How many square pieces of largest size can be cut from the plank, if no wastage is allowed ?
- What is the HCF of \(x^4 - 13x^2y^2 - 300y^4\), \(x^3 - 4x^2y - 4xy^2 - 5y^3\) and \(x^3 - 125y^3\)?
- Let \(N\) be a 5-digit number. When \(N\) is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of \(N\)?
Let \(N\) be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of he following is correct?
Let \(n\) be a natural number. The HCF of \(n, n + 10\) is 10. If the LCM is \(x\) (a 2-digit number), then how many values of \(x\) are possible?
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?
What is the least number which when divided by 12,20 and 24 leaves in each case a remainder of 8?
Which of the following is a pair of co-primes?
