Solved on Dec 15, 2023

Find the least common multiple of 8 and 10 seconds to determine when two fireflies flashing every 8 and 10 seconds, respectively, will flash together again.

STEP 1

Assumptions
1. The first firefly flashes every 8 seconds.
2. The second firefly flashes every 10 seconds.
3. We are looking for the smallest positive time in seconds when both fireflies will flash together again after they have just flashed simultaneously.

STEP 2

To find when both fireflies will flash together again, we need to find the least common multiple (LCM) of their individual flashing intervals.

STEP 3

List the prime factors of each flashing interval.
For 8 seconds: $8 = 2^3$ For 10 seconds: $10 = 2 \times 5$

STEP 4

Identify the highest powers of all prime factors present in both numbers.
For 2: The highest power is $2^3$ (from 8 seconds interval). For 5: The highest power is $5$ (from 10 seconds interval).

STEP 5

Multiply these highest powers to find the LCM.
$LCM = 2^3 \times 5$

STEP 6

Calculate the LCM.
$LCM = 8 \times 5 = 40$

STEP 7

The LCM of 8 and 10 is 40. Therefore, both fireflies will flash together again after 40 seconds.
Both fireflies will flash at the same time again after 40 seconds.

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Find the least common multiple of 8 and 10 seconds to determine when two fireflies flashing every 8 and 10 seconds, respectively, will flash together again.

STEP 1

Assumptions
1. The first firefly flashes every 8 seconds.
2. The second firefly flashes every 10 seconds.
3. We are looking for the smallest positive time in seconds when both fireflies will flash together again after they have just flashed simultaneously.

STEP 2

To find when both fireflies will flash together again, we need to find the least common multiple (LCM) of their individual flashing intervals.

STEP 3

List the prime factors of each flashing interval.
For 8 seconds: $8 = 2^3$ For 10 seconds: $10 = 2 \times 5$

STEP 4

Identify the highest powers of all prime factors present in both numbers.
For 2: The highest power is $2^3$ (from 8 seconds interval). For 5: The highest power is $5$ (from 10 seconds interval).

STEP 5

Multiply these highest powers to find the LCM.
$LCM = 2^3 \times 5$

STEP 6

Calculate the LCM.
$LCM = 8 \times 5 = 40$

STEP 7

The LCM of 8 and 10 is 40. Therefore, both fireflies will flash together again after 40 seconds.
Both fireflies will flash at the same time again after 40 seconds.

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Find the least common multiple of 8 and 10 seconds to determine when two fireflies flashing every 8 and 10 seconds, respectively, will flash together again.

STEP 1

Assumptions1. The first firefly flashes every 8 seconds.2. The second firefly flashes every 10 seconds.3. We are looking for the smallest positive time in seconds when both fireflies will flash together again after they have just flashed simultaneously.

STEP 2

To find when both fireflies will flash together again, we need to find the least common multiple (LCM) of their individual flashing intervals.

STEP 3

List the prime factors of each flashing interval.For 8 seconds: 8=238 = 2^38=23 For 10 seconds: 10=2×510 = 2 \times 510=2×5

STEP 4

Identify the highest powers of all prime factors present in both numbers.For 2: The highest power is 232^323 (from 8 seconds interval). For 5: The highest power is 555 (from 10 seconds interval).

STEP 5

Multiply these highest powers to find the LCM.LCM=23×5LCM = 2^3 \times 5LCM=23×5

STEP 6

Calculate the LCM.LCM=8×5=40LCM = 8 \times 5 = 40LCM=8×5=40

STEP 7

The LCM of 8 and 10 is 40. Therefore, both fireflies will flash together again after 40 seconds.Both fireflies will flash at the same time again after 40 seconds.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Assumptions
1. The first firefly flashes every 8 seconds.
2. The second firefly flashes every 10 seconds.
3. We are looking for the smallest positive time in seconds when both fireflies will flash together again after they have just flashed simultaneously.

List the prime factors of each flashing interval.
For 8 seconds: $8 = 2^3$ For 10 seconds: $10 = 2 \times 5$

Identify the highest powers of all prime factors present in both numbers.
For 2: The highest power is $2^3$ (from 8 seconds interval). For 5: The highest power is $5$ (from 10 seconds interval).

Multiply these highest powers to find the LCM.
$LCM = 2^3 \times 5$

Calculate the LCM.
$LCM = 8 \times 5 = 40$

The LCM of 8 and 10 is 40. Therefore, both fireflies will flash together again after 40 seconds.
Both fireflies will flash at the same time again after 40 seconds.