3 lights blink in 30 seconds, 45 seconds, and 90 seconds. How many times they will
blink together in 30 min?

To find out how many times the three lights will blink together in 30 minutes, we first need to determine the least common multiple (LCM) of the three blink intervals: 30 seconds, 45 seconds, and 90 seconds.

### Step 1: Find the LCM of 30, 45, and 90 seconds.
- The prime factorization of 30 is: \( 30 = 2 \times 3 \times 5 \)
- The prime factorization of 45 is: \( 45 = 3^2 \times 5 \)
- The prime factorization of 90 is: \( 90 = 2 \times 3^2 \times 5 \)

To find the LCM, we take the highest powers of all prime factors involved:
- For \( 2 \), the highest power is \( 2^1 \) (from 30 and 90).
- For \( 3 \), the highest power is \( 3^2 \) (from 45 and 90).
- For \( 5 \), the highest power is \( 5^1 \) (from all three).

Thus, the LCM is:
\[
\text{LCM} = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90 \, \text{seconds}.
\]

### Step 2: Convert 30 minutes to seconds.
\[
30 \, \text{minutes} = 30 \times 60 = 1800 \, \text{seconds}.
\]

### Step 3: Determine how many times they blink together.
Now, we divide the total time by the LCM to find out how many times they blink together:
\[
\frac{1800}{90} = 20.
\]

Therefore, the three lights will blink together **20 times** in 30 minutes.

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Do you want more details or have any questions?

Here are some related questions for further exploration:
1. How can you calculate the LCM of more than three numbers?
2. What is the greatest common divisor (GCD) of 30, 45, and 90?
3. How many times would the lights blink together in one hour?
4. How often would they blink together if the blink intervals were prime numbers?
5. How can you use LCM and GCD in other real-world problems like scheduling?

**Tip**: Understanding LCM helps in solving synchronization problems like coordinating schedules or cyclic events!

3 lights blink in 30 seconds, 45 seconds, and 90 seconds. How many times they will blink together in 30 min?

Learn how to calculate how many times 3 lights with blink intervals of 30, 45, and 90 seconds will blink together in 30 minutes. This problem covers the use of LCM and prime factorization for synchronization, suitable for students in Grades 6-8.

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