What is the shortest lenght of television cable that culd be cut into either a whole number of 18ft pieces or a whole number of 30ft pieces?
hint:
multiples of 18
18 36 54 72 90 108 ...
multiples of 30
30 60 90 120 ...
What do you think ?
To find the shortest length of television cable that can be cut into either a whole number of 18ft pieces or a whole number of 30ft pieces, we need to find the least common multiple (LCM) of 18 and 30.
Step 1: Find the prime factors of each number:
- The prime factors of 18 are 2, 3, and 3.
- The prime factors of 30 are 2, 3, and 5.
Step 2: Determine the highest power of each prime factor that appears in either number:
- The highest power of 2 in either number is 2.
- The highest power of 3 in either number is 3.
- The highest power of 5 in either number is 1.
Step 3: Multiply the prime factors using the highest powers determined in step 2:
2^2 * 3^3 * 5^1 = 180
Therefore, the LCM of 18 and 30 is 180.
Hence, the shortest length of television cable that can be cut into either a whole number of 18ft pieces or a whole number of 30ft pieces is 180ft.
To find the shortest length of television cable that can be cut into either a whole number of 18ft pieces or a whole number of 30ft pieces, we need to find the least common multiple (LCM) of 18 and 30.
One way to find the LCM is by listing the multiples of both numbers until we find a common multiple. Alternatively, we can use a mathematical approach by finding the product of the numbers and dividing it by their greatest common divisor (GCD).
Let's use the second approach:
Step 1: Calculate the GCD of 18 and 30.
The factors of 18 are: 1, 2, 3, 6, 9, and 18.
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, and 30.
The common factors are: 1, 2, 3, and 6.
Therefore, the GCD of 18 and 30 is 6.
Step 2: Calculate the LCM using the formula: LCM = (a * b) / GCD(a, b).
a = 18
b = 30
LCM = (18 * 30) / 6
LCM = 540 / 6
LCM = 90
Therefore, the shortest length of television cable that can be cut into either a whole number of 18ft pieces or a whole number of 30ft pieces is 90ft.