Lowest Common Denominator question?
I don’t understand how the answer is 168 for the LCD here. It’s 1/6+5/8+4/7 I got 84 as the LCD by crossing out duplicates and multiplying the prime numbers.

The lowest common denominator (LCD) for the fractions 6 1 ​ , 8 5 ​ , 7 4 ​ is 168. This is found by taking the prime factors of each denominator and their highest powers. The calculation shows 168 is the correct LCD as it includes all unique prime factors.
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Answered by jimthompson5910 | 2025-07-07

The denominators are 6, 8, and 7.
Let's look at the prime factorization of each denominator.

6 = 2*3
8 = 2 2 2
7 = 7

Since 7 is already prime, there's not much we can do with it in terms of factoring. The unique primes that show up are: 2, 3, 7

2 shows up at most three times, so 2^3 is part of the LCD.
3 shows up once, so 3^1 = 3 is also part of the LCD.
7 shows up once so 7^1 = 7 is the last part of the LCD.

We multiply all of those pieces together to get: 2^3 3 7 = 8 3 7 = 24*7 = 168 is the LCD

Here's another approach.
Focus on the first two denominators 6 and 8 only. We'll revisit 7 later.
To get the LCM of two values, multiply them together. Then divide by their GCF.
So,
LCM = 6*8/(GCF of 6 and 8)
LCM = 48/2
LCM = 24
The LCM of 6 and 8 is 24. You can confirm this by listing the multiples of each.
We'll finally revisit 7 again. The original set of denominators was {6,8,7}. Replace the "6,8" with 24 since it's the LCM of 6 and 8.
We go from {6,8,7} to {24,7}
Then repeat the process to get
LCM = 24*7/GCF
LCM = 24*7/1
LCM = 168
This approach is a bit slower if you have more than 3 values in the set, so it's probably more efficient to follow the first method.

Answered by jimthompson5910 | 2025-07-07