The greatest common factor of 2 numbers is
14, the least common multiple is 168. If the 2 numbers are not 14 and 168, what is the sum?
1st number : 14a = 2(7)(a)
2nd number : 14b = 2(7)(b)
168 = 2(2)(2)(3)(7)
14ab = 8(3)(7)
ab = 12
could be:
a b
1 12
2 6
3 4
If a=1, b=12 , the 2 numbers are 14 and 168 , but we are told that's not it
if a = 2, b = 6, the 2 numbers are 28 84
check:
LCF of 28 and 84 would be 28
if a = 3, b = 4
the two numbers are 42 and 56
LCF = 14
42 = 2x3x7
56 = 2x2x2x7
LCM = 2x2x2x3x7 = 168 , YEAHH
so the sum of the two numbers is 42+56 = 98
thank you a bunch reiny
To find the sum of the two numbers, we need to determine what those numbers are first. We are given that the greatest common factor (GCF) of the two numbers is 14, and the least common multiple (LCM) is 168.
Let's work step by step to find the two numbers.
Step 1: Finding the prime factors
To determine the GCF and LCM, we need to find the prime factorization of 168.
168 can be broken down into its prime factors as follows:
168 = 2 * 2 * 2 * 3 * 7
Step 2: Finding the GCF
The GCF is the largest factor that the two numbers share. Since the GCF is 14, we need to find the prime factorization of 14.
14 = 2 * 7
Step 3: Finding the LCM
The LCM is the smallest multiple that is common to both numbers. We have already found the prime factorization of 168, so we can proceed with finding the LCM.
To obtain the LCM, we include all the prime factors of both numbers, along with their highest powers.
Prime factorization of 168: 2^3 * 3^1 * 7^1
Since the GCF is 14 = 2 * 7, we remove the common factors once and include what is left from the LCM.
Therefore, the LCM = 2^3 * 3^1 * 7^1 = 8 * 3 * 7 = 168.
Now that we know the prime factorization of both numbers, we can determine the two numbers.
Step 4: Finding the two numbers
We take the prime factors of 168 and divide them into two groups: one group that contains all the prime factors from the GCF and another group with the remaining prime factors from the LCM.
Group 1 with GCF factors: 2 * 7 = 14
Group 2 with additional LCM factors: 2 * 2 * 3 * 7 = 84
Therefore, the two numbers are 14 and 84.
Step 5: Finding the sum
Since the question states that the numbers are not 14 and 168, the sum of the two numbers is 14 + 84 = 98.
So, the sum of the two numbers is 98.