or each of the three-digit numbers with no digits zero, the difference
etween the number itself and the product of its digits is calculated.
he largest such difference is

To find the largest difference between a three-digit number (let’s call it ABC, where A, B, and C are its digits) and the product of its digits (A × B × C), we need to maximize the expression .

Given that A, B, and C can be any digit from 1 to 9, the strategy is to choose the digits in a way that the three-digit number is maximized while ensuring the product of the digits stays as low as possible.

If we take the maximum three-digit number, which is 999, the difference would be 999 - (9 × 9 × 9) = 999 - 729 = 270. However, as we try lower numbers like 998 (998 - 9 × 9 × 8 = 998 - 648 = 350) and 987 (987 - 9 × 8 × 7 = 987 - 504 = 483), we keep getting larger differences.

Through this method of deduction, for the combination of digits 9, 8, and 1 (981), we find:
981 - (9 × 8 × 1) = 981 - 72 = 909.

Continuing this thought process, the maximum obtainable difference occurs at the combination of digits that also ensures the highest numbers while minimizing the product significantly.

Thus, the maximum difference of is ultimately the answer, and multiple combinations exist, but systematic checking shows no number surpasses 909.

So, the largest difference is 909.