1. The prime factorization of two numbers, a and b, have some missing prime factors. Show/explain your work for full marks. [ /4 marks]
A=2x3x?
B=?x?
a. The GCF of a and b is 5. What is the value of A?
b. The LCM of a and b is 210. What is the value of B?
a. Since the GCF of a and b is 5, it means that the missing prime factor in A must be 5. So, the value of A is 2x3x5.
b. The prime factorization of 210 is 2x3x5x7. To find the missing prime factors in B, we need to divide 210 by A, which is 2x3x5. The result is B = 7.
Therefore, the value of A is 2x3x5 and the value of B is 7.
a. The GCF (Greatest Common Factor) of a and b is 5. This means that both a and b must have 5 as a factor. In the given prime factorization of A, there is no prime factor 5 listed. So, A must be equal to 2x3x5.
b. The LCM (Least Common Multiple) of a and b is 210. To find the value of B, we need to find the missing prime factors that are present in the LCM but not in A.
First, let's factorize 210:
210 = 2 x 3 x 5 x 7
In the given prime factorization of A, we already have 2, 3, and 5. However, we don't have prime factor 7.
Therefore, the missing prime factors of B are 7. Hence, B must be equal to 7x?.
So, the value of B is 7x?.
To find the missing prime factors in the prime factorization of numbers A and B, we'll start by using the given information.
a. The GCF (Greatest Common Factor) of A and B is 5. This means that both A and B must have a factor of 5, as the GCF is the largest number that divides both A and B evenly.
For A, we know that the given prime factors are 2 and 3:
A = 2 × 3 × ?
Since A must have a factor of 5, one of the missing prime factors is 5. Therefore, the prime factorization of A becomes:
A = 2 × 3 × 5
b. The LCM (Least Common Multiple) of A and B is 210. This means that 210 is the smallest number that can be divided evenly by both A and B.
We already found the prime factorization of A as:
A = 2 × 3 × 5
To find the prime factorization of B, we need to determine the remaining prime factors. Since A and B have a common factor of 5 (as indicated by the GCF being 5), B cannot have a factor of 5.
We know that the LCM of A and B is 210, so the prime factorization of B must include the prime factors that are not present in A: 2 and 3.
Thus, the prime factorization of B can be written as:
B = 2 × 3 × ?
Now, we need to determine the remaining prime factor(s) that complete the factorization.
We know that the LCM of A and B is 210. Therefore, B must have a factor of 7 since 210 (2 × 3 × 5 × 7) is divisible by 7.
Thus, the complete prime factorization of B becomes:
B = 2 × 3 × 7
To summarize:
a. A = 2 × 3 × 5
b. B = 2 × 3 × 7