Chong Ching Tong from River Valley High School, Singapore and Andrei Lazanu, age12, School: No. 205 Bucharest, Romania approached this problem in different ways.

Here is Chong's working: $$8778 \times 2 = 17556\\ \sqrt{17556} \cong 132.5\\ \frac {(132 \times 133)}{2} = 8778\\ 10296 \times 2 = 20592\\ \sqrt{20592} \cong 143.5\\ \frac {(143 \times 144)}{2} = 10296 \\ 13530 \times 2 = 27060\\ \sqrt{27060} \cong 164.5\\ \frac {(164 \times 165)}{2} = 13530\\ (8778)^2 + (10296)^2 = (13530)^2$$

Here is Andrei's solution:

First I demonstrate that 8778, 10296 and 13530 are triangular numbers, i.e. they can be written in the form n(n+1)/2. In order to do this I decomposed the product of each of the three numbers by 2 in the hope to put it in the form n(n+1)/2. I found:

8778 * 2 = 2 ² * 3 * 7 *11 *19 = 132 * 133.

So,

8778 =

132 ×133

and so 8778 is a triangular number.

10296 * 2 = 2 ² * 11 * 13 = 143 * 144.

So,

0296 =

143 ×144

, and it is a triangular number.

13530 * 2 = 2 ² * 3 * 5 * 11 * 41 = 164 * 165.

So,

3530 =

164 ×165

is also a triangular number.

Now, I demonstrate that the three numbers are a Pythagorean triple. The greatest number is 13530.

13530 ² = 183060900

8778 ² + 10296 ² = 77053284 + 106007616 = 183060900

So, the three numbers are a Pythagorean triple