N and P are integers greater than 1
5N is the square of a number
75NP is the cube of a number
the smallest value for N+P IS what?
i cant figure it out, please help
thank you so much
5N is a square: squares: 4,9,16,25,36,49,64,81,100,...
Now which of those are divisible by five?
now look at cubes. 8,27,64,125,256,...15^3
which of those are divisible by 75?
so if N is 5, and 75*5*P=15^3
then P is 9
thank you so much
To find the smallest value for N + P, we need to consider the given conditions.
First, we know that 5N is the square of a number. Let's assume that number is x. So, we have the equation:
5N = x^2 ---- (Equation 1)
Next, we know that 75NP is the cube of a number. Let's assume that number is y. So, we have the equation:
75NP = y^3 ---- (Equation 2)
Now, let's consider the prime factorization of the numbers in Equations 1 and 2.
For Equation 1:
5N = x^2
Here, 5 is a prime number, so N must also contain a factor of 5. Let's say N = 5n, where n is an integer.
Then, the equation becomes:
5*(5n) = x^2
25n = x^2
So, x must also have a factor of 5. Let's say x = 5m, where m is an integer.
Now, the equation becomes:
25n = (5m)^2
25n = 25m^2
n = m^2
For Equation 2:
75NP = y^3
Here, 75 is divisible by 5 and 3. So, both N and P must have at least one factor of 5 and one factor of 3.
Since we want to find the smallest value for N + P, we need to minimize N and P.
The smallest values for N and P would be N = 5 and P = 3.
Therefore, the smallest value for N + P is 5 + 3 = 8.
To find the smallest value for N+P in this scenario, we need to analyze the given conditions and identify any patterns or relationships that can help us.
First, let's consider the statement "5N is the square of a number." This means that the product of 5 and N must be a perfect square. We can write this as 5N = k^2, where k is a positive integer.
Next, let's look at the statement "75NP is the cube of a number." This means that the product of 75, N, and P must be a perfect cube. We can express this as 75NP = m^3, where m is a positive integer.
Now, let's see if we can find any common factors or relationships between these equations.
Dividing the equation 75NP = m^3 by 5N = k^2, we get (75NP) / (5N) = (m^3) / (k^2). Simplifying this, we have 15P / N = (m / k)^2.
Since N, P, and (m / k) are all positive integers greater than 1, we can conclude that 15P / N must be a perfect square.
To find the smallest value for N+P, we need to minimize both N and P while maintaining the condition that 15P / N is a perfect square.
Let's start by considering the prime factorization of 15, which is 3 * 5. Since both P and N are integers greater than 1, P must be at least 3, and N must be at least 5.
If we set P = 3 and N = 5, then 15P / N = 15 * 3 / 5 = 9.
Therefore, in this case, the smallest value for N+P is 5 + 3 = 8.
In conclusion, the smallest value for N+P in this scenario is 8, where N = 5 and P = 3.