when will the lcm of two numbers be the product of the number.

Maths

When the GCF of two numbers is 1,the LCM is equal to the product of the two numbers

When the two numbers are coprime, which means they have no common factors other than 1. In that case, the LCM of the two numbers will indeed be the product of the numbers. It's like a beautiful love story where no one interferes, not even their factors!

To find out when the least common multiple (LCM) of two numbers will be equal to the product of the numbers, you need to consider the relationship between the numbers and the factors they have in common.

Let's suppose our two numbers are a and b, and their LCM is L. To understand when L will be equal to the product of a and b, we need to look at the prime factorizations of a and b.

1. Prime factorize both numbers (a and b) into their unique prime factors.
2. Take note of all the unique prime factors from both numbers and their respective highest powers.
3. Multiply these factors together.
4. If the result is equal to a times b, then the LCM is the product of the numbers.

Let's illustrate this with an example:
Suppose we have a = 12 and b = 15.

1. Prime factorize 12: 2^2 * 3^1
2. Prime factorize 15: 3^1 * 5^1
3. Take note of the unique prime factors and their highest powers: 2^2, 3^1, 5^1.
4. Multiply these together: 2^2 * 3^1 * 5^1 = 60.
5. Compare this product to a times b: 12 * 15 = 180.
Since 60 is not equal to 180, the LCM of 12 and 15 is not equal to their product.

Therefore, the LCM of two numbers will only be equal to the product of the numbers when the unique prime factors of each number are different, or when one number is a multiple of the other.

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