The diffrence of the squares of two natural number is 45. The square of smaller number is 4 times the larger number . Find the number.

Let larger number be x and smaller number be y.

From the given conditions. 

y² = 4x                ... (1)

and

x² - y² = 45

Using (1), we have:

x² - 4x - 45 = 0

(x - 9)(x + 5) = 0

x = 9 or x = -5

Since, x is a natural number so it cannot be negative.

Therefore, x = 9

From (1): y² = 36

This gives y = 6 (as y cannot be negative)

 

Thus, the two numbers are 9 and 6.