What Is the LCD of 84 and 126?
The least common denominator of 84 and 126 is 252. It is the smallest number that both denominators divide into evenly, which is the same number as the least common multiple of 84 and 126.
Method 1: List the Multiples
- Multiples of 84: 84, 168, 252
- Multiples of 126: 126, 252
The first number in both lists is 252, so the LCD is 252.
Method 2: Use Prime Factorization
- 84 = 2² × 3 × 7
- 126 = 2 × 3² × 7
Take every prime at its highest power: 2² × 3² × 7 = 252.
Method 3: Use the GCF
The GCF of 84 and 126 is 42, so the LCD is the product divided by the GCF:
(84 × 126) ÷ 42 = 10,584 ÷ 42 = 252
Rewrite the Fractions over 252
Multiply the top and bottom of each fraction so the denominator becomes 252:
- 1/84 = (1 × 3)/(84 × 3) = 3/252
- 1/126 = (1 × 2)/(126 × 2) = 2/252
Now the denominators match, so you can add or compare them: 1/84 + 1/126 = 5/252.
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