Find all natural numbers that satisfy the equation: x^2+y^2=72?

Answer 1

The nice thing about questions like this is we can quickly get an upper bound on the possible values of x and y. We're only looking for whole number solutions and we know y² ≥ 0 for any natural number y (of course, the same applies to x²). This tells us that x² = x² + 0 ≤ x² + y² = 72 (because 0 ≤ y²) which simply says that x ≤ √72 < 9. Now we can just test these values of x (of course, these bounds also hold for y). The only other thing to keep in mind is that we can exploit symmetry in this problem, i.e. if (a, b) is a solution (that is, x = a, y = b, which is to say a² + b² = 72), then it is also the case that (b, a) is a solution (that is, x = b, y= a).

I hope this helps!

Answer 2

First, there are some definitions and characteristics that make this problem very easy to solve.

Natural number: the positive integers (whole numbers) 1, 2, 3, etc., and sometimes zero as well.

The square of a natural number is n•n -- multiplication of the natural number n with itself. It makes a perfect square. A perfect square is also a natural number.

The sum of two natural numbers is also a natural number [note: The difference may not be a natural number because that effectively is adding a natural number with a negative number.]

For the sum of two natural numbers (the squares of natural numbers) to equal 72, each natural number must be less than 72 (if we consider the natural numbers to begin with 1).

   This means that both x² and y² must be less than 72. Well, of course, that means that x and y can only be either 1, 2, 3, 4, 5, 6, 7, or 8 because 9² = 81 (which is greater than 72)..

Now if we have a list of the squares of the numbers from 1 to 8 we merely select the two that produce 72 when added. This is a very simple addition table:

x and y	1	4	9	16	25	36	49	64
1	2	5	10	17	26	37	50	65
4	5	8	13	20	29	40	53	68
9	10	13	18	25	34	45	58
16	17	20	25	32	41	52	65
25	26	29	34	41	50	61
36	37	40	45	52	61	72
49	50	53	58	65
64	65	68

There are actually two answers:

1)  x = 6 and  y = 6

2)  x = 6 and  y = 6

Check:

Does 6² + 6² = 72 ?

36 + 36 = 72 ?

72 = 72 ?yes