What is the LCM of 64/72 and 96?

General
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What is the LCM of 64/72 and 96?

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What is the LCM of 64/72 and 96?

To find the Least Common Multiple (LCM) of two rational numbers such as \frac{64}{72} and an integer like 96, we need to follow a systematic approach. The Least Common Multiple of two numbers is the smallest positive integer that is a multiple of both numbers.

Step-by-Step Procedure

  1. Convert the Rational Number:

    • The first number is \frac{64}{72}, which is a fraction. We can simplify this fraction before processing it further:
    • Simplifying \frac{64}{72} involves finding the greatest common divisor (GCD) of 64 and 72. The GCD of 64 and 72 is 8.
    • Divide both the numerator and the denominator by their GCD:
    \frac{64}{72} = \frac{64 \div 8}{72 \div 8} = \frac{8}{9}

  2. Deal with Integer:

    • The second number is 96, and it is already an integer, so no simplification is necessary.

  3. Consider the LCM of a Fraction and an Integer:

    • When considering the LCM of a rational number \frac{a}{b} and an integer c, it involves finding a common multiple that works for both:
    • The formula for this operation is given by:
    \text{LCM}\left(\frac{a}{b}, c\right) = \frac{\text{LCM}(a, bc)}{\text{GCD}(b, c)}

  4. Apply the Formula:

    • We have a = 8, b = 9, and c = 96.

  5. Calculate LCM(a, bc):

    • First, calculate bc = 9 \times 96 = 864.

    • Next, find the LCM of 8 and 864. To do this, we use the prime factorization method:

      • Prime Factorizations:

        • 8 = 2^3
        • 864 = 2^5 \times 3^3

      • Display the largest power of each prime:

        • 2^5 (from 864)
        • 3^3 (from 864)

      • LCM of 8 and 864 is:

      \text{LCM}(8, 864) = 2^5 \times 3^3 = 864

  6. Calculate GCD(b, c):

    • Find the GCD of 9 and 96:

      • Prime Factorizations:

        • 9 = 3^2
        • 96 = 2^5 \times 3^1

      • The smallest power for each common prime:

        • 3^1 (common prime factor)

      • GCD is:

      \text{GCD}(9, 96) = 3

  7. Final Calculation:

    • Place the results into the LCM formula:
    \text{LCM}\left(\frac{8}{9}, 96\right) = \frac{864}{3} = 288

The Least Common Multiple of \frac{64}{72} and 96 is therefore 288.

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