I am the greatest common factor of three numbers. Their sum is 36. The largest number is twice the smallest and the middle number is the mean of the other two. What are the three numbers? What number am I?
let the 3 numbers be a , b, and c, in order of size, c is the largest
a+b+c = 36
c = 2a
b = (a+c)/2
= (a+2a)/2 = 3a/3
back into the first
a+b+c = 36
a + 3a/2 + 2a = 36
times 2
2a + 3a + 4a = 72
9a = 72
a = 8, then c = 16, and b = 12
Our 3 numbers are 8, 12, and 16
(all informations checks)
the greatest common factor of those is 4
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Let's assume the smallest number is x.
According to the given information, the largest number is twice the smallest number.
So, the largest number = 2x.
The middle number is the mean of the other two numbers (smallest and largest).
So, the middle number = (x + 2x) / 2 = 3x / 2.
The sum of the three numbers is 36.
x + 2x + 3x / 2 = 36.
Adding the x's, we get:
6x / 2 + 3x / 2 = 36.
Combining like terms, we get:
9x / 2 = 36.
To find x, we'll multiply both sides of the equation by 2/9:
(2/9) * (9x / 2) = (2/9) * 36.
x = 8.
Therefore, the smallest number is 8.
The largest number is twice the smallest number:
2x = 2 * 8 = 16.
So, the largest number is 16.
The middle number is the mean of the smallest and largest numbers:
3x / 2 = (3 * 8) / 2 = 24 / 2 = 12.
Therefore, the three numbers are 8, 12, and 16.
As the greatest common factor of the three numbers, you are 8.
To find the three numbers, let's start by assigning variables to each number.
Let the smallest number be x. Since the largest number is twice the smallest, the largest number would be 2x.
And since the middle number is the mean (average) of the other two, we can set up an equation using the sum of the three numbers, which is given as 36.
So, we have:
x + (2x) + (average of x and 2x) = 36
To find the average, add x and 2x, and divide by 2:
(x + 2x) / 2 = (3x / 2)
Now, we can rewrite the equation:
x + 2x + (3x / 2) = 36
To simplify the equation, let's get rid of the fraction by multiplying everything by 2:
2(x + 2x) + 3x = 72
Simplifying further:
2x + 4x + 3x = 72
9x = 72
Dividing both sides of the equation by 9, we find:
x = 8
Therefore, the smallest number is 8, the largest number is 2x, which is 2 * 8 = 16, and the middle number is the average of 8 and 16, which is (8 + 16) / 2 = 12.
Therefore, the three numbers are 8, 12, and 16. And since you are the greatest common factor of these three numbers, you are the number 4 (since 4 is the largest number that can divide all three numbers evenly).