I have a similar problem The LCM of two numbers is 36. The GCF is 6. One number is 6 more than the other. The numbers are ------- and ------. If someone could show me how to do this one, I could figure out the other 5 I have to do (maybe) Thank you very much.
LCM(a,b) = a*b/GCF(a,b)
so,
36 = a(a+6)/6
a^2 + 6a - 216 = 0
(a+18)(a-12) = 0
a = 12
So, the two numbers are 12 and 18
Put 42/48 in simplest form.
To find the two numbers given the LCM, GCF, and the fact that one number is 6 more than the other, we can follow these steps:
Step 1: Identify the GCF and the LCM
The GCF (Greatest Common Factor) is given as 6, and the LCM (Least Common Multiple) is given as 36.
Step 2: Relationship between LCM and GCF
The relationship between the LCM and GCF of two numbers is that their product is equal to the product of the two numbers. In mathematical terms:
LCM × GCF = Number 1 × Number 2
Plugging in the given values, we have:
36 × 6 = Number 1 × Number 2
216 = Number 1 × Number 2
Step 3: Set up equations
Since we are also given that one number is 6 more than the other, we can set up equations using this information.
Let's say Number 1 is x. Therefore, Number 2 would be (x + 6) because it is 6 more than the first number.
So our equation becomes:
216 = x × (x + 6)
Step 4: Solve the equation
Now we need to solve this quadratic equation for x.
Expanding the equation:
216 = x² + 6x
Rearranging the equation:
x² + 6x - 216 = 0
Factoring the equation is one possible method, but in this case, let's solve it by using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values:
x = (-(6) ± √((6)² - 4(1)(-216))) / (2(1))
x = (-6 ± √(36 + 864)) / 2
x = (-6 ± √900) / 2
Taking the square root:
x = (-6 ± 30) / 2
Simplifying:
x = (24 / 2) or x = (-36 / 2)
x = 12 or x = -18
Step 5: Substitute the values back
We found two possible values for x, so we need to substitute them back into our equation to find the actual numbers.
If x = 12:
Number 1 = x = 12
Number 2 = x + 6 = 12 + 6 = 18
If x = -18:
Number 1 = x = -18
Number 2 = x + 6 = -18 + 6 = -12
Therefore, the two numbers are:
12 and 18
or
-18 and -12
Remember, negative numbers are also possible solutions when dealing with mathematics.