How can we express the fractions 34/716 and 5/8 with a common denominator?

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To express the fractions 34/716 and 5/8 with a common denominator, we first need to determine the least common denominator (LCD) of the two fractions.

Step 1: Simplifying 34/716

  • First, we need to simplify 34/716. We can find the greatest common divisor (GCD) of 34 and 716.
  • 34 can be factored into 2 and 17, and 716 can be divided by 2 (since it is even) to get 358, and again to get 179, which is a prime number.
  • Thus, the GCD of 34 and 716 is 34.
  • Dividing both the numerator and the denominator by the GCD:

34/716 = (34 ÷ 34)/(716 ÷ 34) = 1/21

Step 2: Finding the LCD of 21 and 8

  • Next, we find the least common multiple (LCM) of the denominators 21 and 8.
  • The prime factorization for 21 is 3 × 7, and for 8 is 2 × 2 × 2.
  • The LCM is found by taking the highest power of each prime number:
  • LCM = 23 × 31 × 71 = 8 × 3 × 7 = 168.

The least common denominator (LCD) is 168.

Step 3: Converting the fractions

  • Now, we will convert both fractions to have this common denominator.
  • For 1/21:
    • Multiply the numerator and denominator by 8:
    • 1/21 = (1 × 8)/(21 × 8) = 8/168
  • For 5/8:
    • Multiply the numerator and denominator by 21:
    • 5/8 = (5 × 21)/(8 × 21) = 105/168

Final Result:

Both fractions expressed with the least common denominator of 168 are:

  • 34/716 = 8/168
  • 5/8 = 105/168

These equivalent fractions can now be used for further calculations or comparisons.

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