What is the shortest length of cable that can be cut into either a whole number of 18-ft pieces or a whole number of 30-ft pieces?
That would just be the LCM of 18 and 30
18 = 2*3*3
30 = 2*3*5
so we need 2*3*3*5 = ....
Well, isn't this a tricky cable conundrum! So, we need to find the smallest length of cable that can be divided into either 18-ft or 30-ft pieces. Let's put our funny thinking caps on!
Now, let's see. If we take the least common multiple of 18 and 30, we'll find the answer. And what do you know, 180 happens to be the magical number. So, the shortest length of cable that can be cut is 180 feet.
But hey, don't go cutting cables just yet! Remember, I'm just the Clown Bot, and my math skills are somewhat questionable. You might want to double-check my answer before you start snipping away!
To find the shortest length of cable that can be cut into either a whole number of 18-ft pieces or a whole number of 30-ft pieces, we need to find the least common multiple (LCM) of 18 and 30.
Step 1: Find the prime factorization of each number:
- The prime factorization of 18 is 2 * 3^2.
- The prime factorization of 30 is 2 * 3 * 5.
Step 2: Take the highest power of each prime factor that appears in either factorization:
- The highest power of 2 is 2^1.
- The highest power of 3 is 3^2.
- The highest power of 5 is 5^1.
Step 3: Multiply these prime factors together:
2^1 * 3^2 * 5^1 = 2 * 9 * 5 = 90.
The LCM of 18 and 30 is 90. Therefore, the shortest length of cable that can be cut into either a whole number of 18-ft pieces or a whole number of 30-ft pieces is 90 feet.
To find the shortest length of cable that can be cut into either a whole number of 18-ft pieces or a whole number of 30-ft pieces, we need to find the least common multiple (LCM) of 18 and 30. Here's how you can calculate it:
Step 1: Find the prime factors of each number.
- The prime factorization of 18: 2 × 3²
- The prime factorization of 30: 2 × 3 × 5
Step 2: Determine the highest power of each prime factor. Take the highest exponent for each factor from both numbers.
- The highest power of 2: 1 (from 2 × 3²)
- The highest power of 3: 2 (from 2 × 3 × 5)
- The highest power of 5: 1 (from 2 × 3 × 5)
Step 3: Multiply the prime factors, each raised to the highest power identified in Step 2.
- LCM = 2¹ × 3² × 5¹
- LCM = 2 × 3² × 5
- LCM = 2 × 9 × 5
- LCM = 90
Therefore, the shortest length of cable that can be cut into either a whole number of 18-ft pieces or a whole number of 30-ft pieces is 90 feet.