Write a conjecture about the LCM of two prime numbers. Test your conjecture.
it is just the product of the two primes, since there are no common factors.
Conjecture: The least common multiple (LCM) of two prime numbers is equal to the product of the two prime numbers.
To test this conjecture, let's consider two prime numbers, 𝑎 and 𝑏. We will calculate their LCM and compare it to the product of 𝑎 and 𝑏.
For example, let's take 𝑎 = 7 and 𝑏 = 11.
The LCM of 7 and 11 can be determined by finding the smallest multiple that is divisible by both numbers. In this case, we can see that 77 is the smallest multiple that is divisible by 7 and 11.
Now, let's calculate the product of 𝑎 and 𝑏, which is 7 × 11 = 77.
Comparing the LCM (77) and the product of 𝑎 and 𝑏 (77), we can see that they are equal.
Since this example confirms our conjecture, it suggests that for any two prime numbers, their LCM will be equal to the product of the two prime numbers.
Conjecture: The least common multiple (LCM) of two prime numbers is equal to their product.
To test this conjecture, we can choose any two prime numbers and calculate their LCM. Let's take the prime numbers 3 and 5 as an example.
To calculate the LCM of 3 and 5, we can use the following steps:
1. List the prime factorization of both numbers:
- The prime factorization of 3 is 3.
- The prime factorization of 5 is 5.
2. Identify all the unique prime factors from both numbers:
- In this case, we have only one unique prime factor: 3 and 5.
3. Multiply all the unique prime factors together:
- Multiplying 3 and 5, we get 15.
4. The result, 15, is the LCM of 3 and 5.
By following these steps, we can see that the LCM of 3 and 5 is indeed their product: 3 * 5 = 15.
To further verify this conjecture, we can repeat this process with multiple pairs of prime numbers and confirm that the LCM is always equal to their product. This consistency will provide additional evidence to support the conjecture that the LCM of any two prime numbers is equal to their product.