The product of two consecutive odd natural numbers is 255255. Find the largest number among them.

Let's denote the two consecutive odd natural numbers as n and n+2.

Given that the product of these two numbers is 255255, we can write the equation as:

n*(n+2) = 255255

Expanding this gives:

n^2 + 2n - 255255 = 0

This is a quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = 2, and c = -255255.

We can solve this equation for n using the quadratic formula, n = [-b ± sqrt(b^2 - 4ac)] / (2a).

Substituting the values of a, b, and c into the formula gives:

n = [-2 ± sqrt((2)^2 - 41(-255255))] / (2*1) n = [-2 ± sqrt(4 + 1021020)] / 2 n = [-2 ± sqrt(1021024)] / 2 n = [-2 ± 1010] / 2

This gives two possible values for n:

n = (1010 - 2) / 2 = 504 n = (-1010 - 2) / 2 = -506

Since n must be a natural number, we discard the negative solution. Therefore, the smaller of the two consecutive odd natural numbers is 504.

However, the problem asks for the larger of the two numbers. Since the numbers are consecutive odd numbers, the larger number is 2 more than the smaller number, or 504 + 2 = 506.

Therefore, the largest number among the two consecutive odd natural numbers is 506.