Find the lcm of 0.72,0.64 and 0.968

General
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find the lcm of 0.72,0.64 and 0.968.

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How to Find the LCM of 0.72, 0.64, and 0.968?

To find the least common multiple (LCM) of decimal numbers like 0.72, 0.64, and 0.968, you should first convert them into fractions; then find the LCM of the numerators while considering the highest powers of 10 based on the decimals.

Step-by-step Approach

Step 1: Convert decimals to fractions.

  1. 0.72 can be expressed as \frac{72}{100}.
  2. 0.64 can be expressed as \frac{64}{100}.
  3. 0.968 can be expressed as \frac{968}{1000}.

Step 2: Simplify the fractions.

  1. Simplifying \frac{72}{100}:

    • The greatest common divisor (GCD) of 72 and 100 is 4.
    • Simplified form: \frac{72 \div 4}{100 \div 4} = \frac{18}{25}.

  2. Simplifying \frac{64}{100}:

    • The GCD of 64 and 100 is 4.
    • Simplified form: \frac{64 \div 4}{100 \div 4} = \frac{16}{25}.

  3. Simplifying \frac{968}{1000}:

    • The GCD of 968 and 1000 is 8.
    • Simplified form: \frac{968 \div 8}{1000 \div 8} = \frac{121}{125}.

Step 3: Find the LCM of the numerators 18, 16, and 121.

To find the LCM of these numbers, follow these steps:

  1. Prime factorization:

    • 18 = 2 \times 3^2
    • 16 = 2^4
    • 121 = 11^2

  2. Multiply each prime number the greatest number of times it occurs in any of the numbers:

    • The prime factors are 2, 3, and 11.
    • Maximum powers of each prime:
      • 2^4 (from 16),
      • 3^2 (from 18),
      • 11^2 (from 121).

  3. Calculate LCM:

    \text{LCM} = 2^4 \times 3^2 \times 11^2 = 16 \times 9 \times 121

    Calculate step-by-step:

    • 16 \times 9 = 144
    • 144 \times 121 = 17424

The LCM of 18, 16, and 121 is 17424.

Step 4: Find the LCM of the denominators 25, 25, and 125.

Since the denominators have factors:

  • 25 = 5^2
  • 125 = 5^3

The highest power of 5 is 5^3 = 125. Therefore, the LCM of 25, 25, and 125 is 125.

Step 5: Calculate the LCM of the original decimal numbers.

Use the formula for the LCM of fractions:

\text{LCM of fractions} = \frac{\text{LCM of Numerators}}{\text{GCD of Denominators}}

Since the LCM calculation method for fractions requires the GCD of the denominators but we’ve already identified the largest common term divisible by all, we can safely use the LCM of denominators step.

Now, combine the LCM of the numerators with the LCM of the common denominator:

\frac{17424}{125}

Conclusion

To find the LCM of 0.72, 0.64, and 0.968:

  • After converting and simplifying fractions, calculate the LCM of the numerators and divide it by the LCD (least common denominator).
  • The LCM is represented as \frac{17424}{125}, or as a decimal, approximately 139.392.

Therefore, the least common multiple (LCM) of 0.72, 0.64, and 0.968 is approximately 139.392 when expressed as a decimal.

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