P, q, r are positive integer numbers. a, b, c are single digit integers from 0 to 9.

p : q : r = 2 : 3 : 6 Let the constant of proportionality be x (rational number).

p = 2x q = 3x r = 6x

p² + q² + r² = 3a6bc

4x²+9x²+36x² = 30,000+a*1,000+600+b*10+c

49 x² = 30,600+1000*a+10*b+c or 3a6bc

Then factors of LHS are :

1, 7, 49, x, 7x, 49x, x², 7 x², 49x²

there are 9 factors.

========================

6bc divides 3a6bc ??

3a6bc is in the range: 39,699 --- 30,600

3a6bc = 3a * 1000 + 6bc

3a6bc / 6bc = 3a*1000/6bc + 1

49 * 25 * 25 = 30625 that is : a = 0, b = 2 and c = 5.

so 6bc divides 3a6bc for the above numbers.

perhaps more numbers too..

================

__The solution:__

the range of 3a6bc is 39699 to 30600.

so range of x² is from 39699/49 = 810.1 to 624.48

so range of x is from: 28 to 25.

so 49 * 625 = 30625 =======> this is right.

49 * 26² = 33124

49 * 27² = 35721

======================

Deductive way of solving this problem.

N = 49 * x² = 3a6bc

if 6bc is a factor of 3a6bc, then 6bc is a multiple of 7, 7², x².

625 = 25² is a perfect square. trying it works.