a and b are positive integers that satisfy 18a=b^3 . What is the minimum possible value of a+b ?
To find the minimum possible value of a+b, we need to determine the values of a and b that satisfy the given equation 18a = b^3.
To start, let's focus on the prime factorization of 18. The prime factors of 18 are 2 and 3, which can be written as 2 * 3 * 3.
Now, let's consider the prime factorization of b^3. Since the left side of the equation 18a = b^3 contains a factor of 2, the prime factorization of b^3 must also include 2. Therefore, b must be even.
Let's rewrite b as 2k, where k is a positive integer.
Substitute b = 2k in the equation 18a = b^3:
18a = (2k)^3
18a = 8k^3
a = (8k^3)/18
a = (4k^3)/9
Now, let's analyze the expression (4k^3)/9. Since k is a positive integer, the minimum value of k that makes (4k^3)/9 an integer is k = 3 (taking the cube of 3 gives us 27, making the denominator a multiple of 9).
Plugging k = 3 back into the expression for a, we get:
a = (4 * (3^3))/9
a = (4 * 27)/9
a = 12
Therefore, the minimum possible value of a is 12 for this equation.
To find the minimum possible value of a+b, we can substitute this value of a into the equation:
a + b = 12 + 2k
Since b = 2k, we know that b is also even. To minimize the value of a+b, we need to find the minimum even value for b, which is b = 2.
Substituting b = 2, we get:
a + b = 12 + 2(2)
a + b = 12 + 4
a + b = 16
So, the minimum possible value of a+b is 16.