Two numbers--
Their greatest common factor is 3.
Their least common multiple is 90.
How many answers are possible? Explain.
There are 4 possible answers to this question:
1) 6 and 24
2) 9 and 30
3) 15 and 18
4) 3 and 90
Since 90*3=270 and 270=3*3*3*2*5.
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Winese
To find the possible pairs of numbers that satisfy these conditions, we need to consider the prime factorization of the least common multiple (LCM) of the two numbers. In this case, the LCM is 90, which can be expressed as the product of its prime factors: 2 * 3 * 3 * 5.
Since the greatest common factor (GCF) of the two numbers is 3, each number must have at least one factor of 3. Furthermore, to achieve an LCM of 90, each number must have all the prime factors of 90, which include 2, 3, and 5.
Since the GCF is 3, one of the numbers must be a multiple of 3 (because it must have at least one factor of 3). However, it cannot be a multiple of 9 (since the GCF is not 9), so we have three possible choices for the first number: 3, 3 * 2 (6) or 3 * 5 (15).
Next, we determine the possible values for the second number. To obtain a GCF of 3 and an LCM of 90, the second number has to be a multiple of 3 and include the prime factors of 90 that are not already accounted for (i.e., 2 and 5). We already know that the second number must be a multiple of 3, so we have the same three choices: 3, 6, or 15.
Given the two possible choices for each number (3 or 6 or 15), there are a total of 3 * 3 = 9 possible pairs of numbers that satisfy the conditions.
In summary, there are 9 possible pairs of numbers that have a greatest common factor of 3 and a least common multiple of 90.
To find how many possible answers exist, we need to determine the prime factorization of 90 and consider the possible combinations of factors.
The prime factorization of 90 is 2 * 3^2 * 5.
Since the greatest common factor is 3, we can represent one number as 3 multiplied by a combination of the remaining factors. Let's consider the possibilities:
1. The other number can be 2 * 3^2 * 5.
2. The other number can be just 2 * 5 (without any 3 factors).
3. The other number can be just 3 (without any 2 or 5 factors).
4. The other number can be 2 * 3 (without any 5 factors).
So, there are four possible combinations for the other number.
Therefore, there are four possible answers.