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Find all Numbers II (Posted on 2022-04-24)

Determine all possible value(s) of a non-leading zero, 5-digit positive integer N that satisfy each of these given conditions:
(i) Product of the digits of N is equal to the sum of its digits.
(ii) N is divisible by the sum of its digits.
(iii) N does NOT contain the digit 3.

Added for completeness:
What is the total number of values of N if we disregard condition (iii)?

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Solution

| Comment 3 of 4 |

Just looking at condition 1, a sum of five digits equals a product of five digits when the digits are {1,1,2,2,2} or {1,1,1,3,3} or {1,1,1,2,5}. When written as 5 digit numbers there are 10+10+20=40 possible numbers N.

Adding condition 2, the set {1,1,2,2,2} can only be arranged as 22112. The set {1,1,1,3,3} has all 10 arrangements valid. And the set {1,1,1,2,5} has no valid arrangements. So there are 11 possible numbers N at this point, answering the bonus question.

Adding condition 3 eliminates all solutions derived from {1,1,1,3,3} leaving a unique solution for N as 22112.

Posted by Brian Smith on 2022-04-24 10:07:27

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