Triangular triples

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:

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: $$8778\times 2=2^2\times 3\times 7\times 11\times 19=132\times 133$$ So, $$8778=\frac{132\times 133}{2}$$ and so 8778 is a triangular number. $$10296\times 2=2^2\times 11\times 13=143\times 144$$ So, $$0296=\frac{143\times 144}{2}$$, and it is a triangular number. $$13530\times 2=2^2\times 3\times 5\times 11\times 41=164\times 165$$ So, $$3530=\frac{164\times 165}{2}$$ is also a triangular number.

Now, I demonstrate that the three numbers are a Pythagorean triple. The greatest number is 13530 $${13530}^2=183060900$$ $${8778}^2+{10296}^2=77053284+106007616=183060900$$