Twenty fully worked least common denominator problems, arranged from simple two-fraction pairs up through three-fraction sums and mixed number conversions. Every card below shows the method used, the key working line, the resulting LCD, and the equivalent fractions rewritten over that shared denominator, so the arithmetic is never left as a bare answer. Together the set rotates through all three standard techniques: the list of multiples method, prime factorization, and the GCF (greatest common factor) method, so each one gets a fair share of practice.
No calculator is needed to read through these. Every step is spelled out in plain arithmetic. When you are ready to work your own fractions, the live calculator on the LCD Calculator homepage handles any combination of fractions, integers, and mixed numbers instantly.
Each card states the problem, names the method, shows the working, and ends with the equivalent fractions and their sum. Read across the grid to see how the same underlying idea, finding the smallest shared denominator, plays out differently depending on the numbers involved.
Because 4 and 6 are both small, the fastest route is simply listing multiples until one repeats in both lists. That first shared value is the LCD.
8 and 12 share the factor 4, so dividing the product of the denominators by their GCF skips straight past a long multiples list.
Whenever one denominator divides evenly into the other, the larger denominator is the LCD outright. No list is even needed once you notice 9 ÷ 3 = 3.
Breaking 6 and 9 into primes shows they both carry a factor of 3. The LCD needs that 3 only once, but at its highest power from either number.
3 and 4 share no common factor, so their multiples never overlap until reaching the product of the two numbers.
6 and 8 share a GCF of 2, so the formula shortcut lands on 24 without needing to write out either multiples list.
4 and 5 are coprime, so the multiples lists have to run all the way to their product, 20, before a match appears.
3 and 7 are both prime and share no factors, so the LCD is their straight product, 21, the same answer the GCF method would give since GCF(3, 7) = 1.
5 is already prime and 6 splits into 2 × 3, so every prime involved, 2, 3, and 5, appears only once across both denominators.
3 and 5 share no common factor, so their GCF is 1 and the formula reduces to a plain multiplication: 3 × 5.
This is the smallest pair on this page, and it shows the list method at its clearest: two short lists, one quick overlap at 6.
4 and 9 are both perfect squares of primes, 2² and 3², so neither shares a factor with the other, and the LCD is simply their product.
8 and 9 are consecutive integers with no shared factor, so GCF(8, 9) = 1 and the LCD climbs all the way to their product, 72.
6 and 10 both contain a factor of 2, so that prime is only counted once at its highest power, alongside the 3 from 6 and the 5 from 10.
9 and 12 share a GCF of 3, which is what keeps the LCD at 36 instead of the full product of 108.
6 and 15 both include a factor of 3, so the prime breakdown keeps that 3 to a single copy while still capturing the 2 from 6 and the 5 from 15.
With three denominators at once, prime factorization scales far better than a triple list of multiples. Every prime just needs its single highest power across all three numbers.
3, 4, and 5 share no common factors with each other, so even the list method has to run each list out to their shared product, 60, before all three align.
Mixed numbers are converted to improper fractions before anything else: 1 1/2 becomes 3/2, and 2 3/4 becomes 11/4. Only then do the denominators, 2 and 4, get compared.
After converting 2 2/3 to 8/3 and 3 1/4 to 13/4, the denominators 3 and 4 share no common prime, so the prime factorization method multiplies 2² by 3 directly.
The twenty problems above rotate through three methods so that whichever one a denominator pair suits best gets demonstrated with real numbers.
Used in Examples 1, 3, 5, 7, 8, 11, 18, and 19. This method lists the multiples of each denominator until the same number shows up in every list. It is the most visual of the three methods and works best when denominators stay under about 20, since the lists stay short. Example 3 (LCD of 3 and 9) shows the special case where one denominator already divides the other, so the larger number is the LCD without any real searching.
Used in Examples 4, 9, 12, 14, 16, 17, and 20. Each denominator is broken into prime numbers, the highest power of every prime that appears is kept, and those powers are multiplied together. This method scales the best of the three, which is why it is the natural choice for the three-fraction problem in Example 17 and the mixed-number problem in Example 20, situations where listing multiples for every number involved would take far longer.
Used in Examples 2, 6, 10, 13, and 15. This method multiplies the two denominators together and divides by their greatest common factor: LCD = (a × b) ÷ GCF(a, b). It is the fastest route when the GCF is obvious at a glance, such as GCF(8, 12) = 4 in Example 2, and it collapses to a simple product whenever the denominators share no common factor at all, as in Examples 10 and 13.
Examples 17 and 18 extend the same ideas to three fractions at once, and Examples 19 and 20 add a conversion step for mixed numbers before either method is applied. In every case the three methods agree. They are different paths to the exact same least common denominator.
Reading worked examples builds the pattern, but the fastest way to check a homework problem or a real-world fraction is to run it through the calculator directly.
The LCD Calculator on the homepage accepts simple fractions, mixed numbers, and whole numbers together, up to five at a time, and returns the least common denominator, the equivalent fractions, and a full prime-factor breakdown instantly, with no rounding and no ads blocking the result.
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