To solve the problem, we follow these steps:

Step 1: Use the algebraic identity

Given three numbers (a, b, c):
[ (a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) ]

We know:

• (a+b+c = 27)
• (a^2 + b^2 + c^2 = 293)

Substitute into the identity:
[ 27^2 = 293 + 2(ab + bc + ca) ]
[ 729 = 293 + 2(ab + bc + ca) ]
[ 2(ab + bc + ca) = 729 - 293 = 436 ]
[ ab + bc + ca = 218 ]

Step 2: Find the three numbers

We need integers (a, b, c) that sum to 27 and have a sum of squares 293.

Testing possible values, we find:
(4, 9, 14) (sum: (4+9+14=27); sum of squares: (16+81+196=293))

Step 3: Calculate the product

[ abc = 4 \times 9 \times 14 = 504 ]

Answer: (\boxed{504})

作者声明:本文包含人工智能生成内容。