jmc

Question

There are positive integers that have these properties:

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I. The sum of the squares of their digits is

50,

and

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II. Each digit is larger than the one on its left.

What is the product of the digits of the largest integer with both properties?

Accepted Answer

To meet the first condition, numbers which sum to 50 must be chosen from the set of squares 1, 4, 9, 16, 25, 36, 49. To meet the second condition, the squares selected must be different. Consequently, there are three possibilities: 1+49, 1+4+9+36, and 9+16+25. These correspond to the integers 17, 1236, and 345, respectively. The largest is 1236, and the product of its digits is 1236=36.