Finding LCM by Listing Multiples

Find the LCM of $3$ and $5$.

List multiples of 3: $3 \times 1 = 3$, $3 \times 2 = 6$, $3 \times 3 = 9$, $3 \times 4 = 12$, $3 \times 5 = 15$, ... = $3, 6, 9, 12, 15, 18, 21, ...$

List multiples of 5: $5 \times 1 = 5$, $5 \times 2 = 10$, $5 \times 3 = 15$, ... = $5, 10, 15, 20, 25, ...$

Find the smallest common multiple: Both lists contain $15$ = $\text{LCM}(3, 5) = 15$

Answer: The LCM of $3$ and $5$ is $15$.

Finding LCM Using Prime Factorization

Find the LCM of $12$ and $18$.

Find prime factorization of 12: $12 = 2 \times 6 = 2 \times 2 \times 3$ = $12 = 2^2 \times 3$

Find prime factorization of 18: $18 = 2 \times 9 = 2 \times 3 \times 3$ = $18 = 2 \times 3^2$

Take highest power of each prime: Highest power of $2$: $2^2$. Highest power of $3$: $3^2$ = $2^2 \times 3^2$

Multiply: $4 \times 9 = 36$ = $\text{LCM}(12, 18) = 36$

Answer: The LCM of $12$ and $18$ is $36$.

LCM of Three Numbers

Find the LCM of $4$, $6$, and $9$.

Find prime factorizations: $4 = 2^2$, $6 = 2 \times 3$, $9 = 3^2$ = Three factorizations

Identify all prime factors: Primes used: $2$ and $3$ = Two prime factors

Take highest power of each: Highest power of $2$: $2^2$ (from 4). Highest power of $3$: $3^2$ (from 9) = $2^2 \times 3^2$

Calculate the LCM: $4 \times 9 = 36$ = $\text{LCM}(4, 6, 9) = 36$

Answer: The LCM of $4$, $6$, and $9$ is $36$.

Mistake: Multiplying the numbers instead of finding LCM

Why: While $4 \times 6 = 24$ gives a common multiple, it is not always the least. The LCM of $4$ and $6$ is $12$, not $24$.

Correct: Always check if there is a smaller common multiple by listing or using prime factorization.

Mistake: Confusing LCM with GCF

Why: LCM is the smallest common multiple; GCF is the largest common factor. For $4$ and $6$: LCM is $12$, GCF is $2$.

Correct: Remember: LCM uses multiples (bigger numbers), GCF uses factors (smaller numbers).

Mistake: Missing a common multiple when listing

Why: Students sometimes stop listing too early and pick a number that is not the smallest.

Correct: Continue listing until you find a match in both lists. The first match is the LCM.

Bus Schedules

Use LCM to figure out when two buses on different schedules will arrive at the same time.

Bus A arrives every $8$ minutes and Bus B arrives every $12$ minutes. If both arrive at 9:00 AM, when will they next arrive together?

Adding Fractions

LCM helps find the common denominator when adding fractions with different denominators.

To add $\frac{1}{4} + \frac{1}{6}$, find $\text{LCM}(4, 6) = 12$, then rewrite as $\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$.

Q: What is the difference between LCM and GCF?

A: LCM (Least Common Multiple) is the smallest number that both numbers divide into evenly. GCF (Greatest Common Factor) is the largest number that divides into both numbers evenly. LCM is always greater than or equal to the larger number; GCF is always less than or equal to the smaller number.

Q: Can the LCM of two numbers equal one of the numbers?

A: Yes! If one number is a multiple of the other, the LCM equals the larger number. For example, $\text{LCM}(3, 6) = 6$ because $6$ is already a multiple of $3$.

Q: Is there a shortcut to find LCM?

A: Yes! You can use the formula: $\text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)}$. For example, for $4$ and $6$: GCF is $2$, so LCM $= \frac{4 \times 6}{2} = \frac{24}{2} = 12$.