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Difficulty: EasyFactors, Multiples, and Prime Factorization

A student is adding the fractions 512\frac{5}{12} and 718\frac{7}{18}. What is the least common denominator they should use to add these fractions?

  1. A
    6
  2. B
    30
  3. 36Answer
  4. D
    72
  5. E
    216

Answer

36
To find the least common denominator of the two fractions, find the least common multiple (LCM) of the denominators, 12 and 18. The prime factorization of 12 is 22×32^2 \times 3, and the prime factorization of 18 is 2×322 \times 3^2. The LCM is found by taking the highest power of each prime factor that appears in either factorization: 22×32=4×9=362^2 \times 3^2 = 4 \times 9 = 36.

Step-by-Step Solution

1
Identify the denominators of the fractions to find their least common denominator.
The denominators are 12 and 18.
The least common denominator of two fractions is equal to the least common multiple (LCM) of their denominators.
2
Find the prime factorization of each denominator.
12=22×312 = 2^2 \times 3 and 18=2×3218 = 2 \times 3^2
Prime factorization helps systematically determine the least common multiple.
3
Calculate the least common multiple by multiplying the highest power of each prime factor present.
22×32=4×9=362^2 \times 3^2 = 4 \times 9 = 36
This yields the smallest positive integer that is divisible by both 12 and 18.

Key Concept

Finding the least common denominator by calculating the least common multiple (LCM) of the denominators.
Estimated Time:45s
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Topics

Factors, Multiples, and Prime Factorization

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