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Data Interview Question

Determine the Value of X

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Answer

To solve the equation 16p+1=x316p + 1 = x^3 where pp is a prime number, we need to find the value of xx. Let's break down the solution step-by-step.

Step 1: Understand the Equation

The given equation is:

16p+1=x316p + 1 = x^3

Rearranging it, we get:

16p=x3116p = x^3 - 1

The expression x31x^3 - 1 can be factored using the identity for the difference of cubes:

x31=(x1)(x2+x+1)x^3 - 1 = (x - 1)(x^2 + x + 1)

Thus, the equation becomes:

16p=(x1)(x2+x+1)16p = (x - 1)(x^2 + x + 1)

Step 2: Analyze the Factors

We know that pp is a prime number, which implies that the product (x1)(x2+x+1)(x - 1)(x^2 + x + 1) divided by 16 should result in a prime number. Let's analyze the factors:

  • x1x - 1 must be a multiple of 16 for pp to remain a prime number.
  • x2+x+1x^2 + x + 1 must be an odd number because the sum of an even number x(x+1)x(x+1) and 1 is always odd.

Step 3: Solve for xx

Given that x1x - 1 must be a multiple of 16, let's assume:

x1=16kx - 1 = 16k

where kk is a positive integer. Therefore, x=16k+1x = 16k + 1.

Substituting back into the equation:

16p=(16k)(x2+x+1)16p = (16k)(x^2 + x + 1)

Since pp is a prime number, kk must be 1 to keep pp prime. Therefore:

x1=16x - 1 = 16 x=16+1=17x = 16 + 1 = 17

Step 4: Verify the Solution

Substituting x=17x = 17 back into the original equation:

16p+1=17316p + 1 = 17^3 16p+1=491316p + 1 = 4913 16p=491216p = 4912 p=491216=307p = \frac{4912}{16} = 307

Check if 307 is a prime number:

  • 307 is not divisible by any prime number up to its square root (approximately 17.5), confirming that 307 is a prime number.

Thus, the solution is consistent, and the value of xx is:

17\boxed{17}