Problem 038


Project Euler

Pandigital multiples

Problem 38

Take the number 192 and multiply it by each of 1, 2, and 3:

192 × 1 = 192
192 × 2 = 384
192 × 3 = 576

By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3)

The same can be achieved by starting with 9 and multiplying by 1, 2, 3, 4, and 5, giving the pandigital, 918273645, which is the concatenated product of 9 and (1,2,3,4,5).

What is the largest 1 to 9 pandigital 9-digit number that can be formed as the concatenated product of an integer with (1,2, … , n) where n > 1?


Since we are looking for the largest value, we will start with the largest possible number to make this work. Notice that there are only 9 digits for the concatenated product, therefore, the result must not exceed 9 digits given x*1 “concatenate” x*2 … x*n. Also, with n>1, therefore, the final concatenate must at least constructed by 1x and 2x. Thus, 9999 is the maximum value allowed and 9 is the minimum value. From here on, using a brute force method of check each number until either one of the digit is used. We can delay the concatenation of the products until a match is found.

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