Math CentralQuandaries & Queries


Question from waiyan, a student:

x ?
using 1 2 3 4 5 6 7 8 9

Hi Waiyan,

There is actually more than one solution to this riddle. I am not going to give you an answer but I can give you some hints to find it.
Using 1 -9, the 4 digit numbers will range from 1234 to 9876. Since the largest product is 9876, the largest 4 digit multiplicand will be 4939 and the smallest can be 1234.

The single digit multiplicand can only be 2, 3, 4 , 6 or 7 for the following reasons:
- any number multiplied by 5 will end in 0 or 5
- any number multiplied by 8 must be smaller than 1232 or the product will be greater than 9876. This is smaller than the smallest possible multiplicand.
- any number multiplied by 9 must be smaller than 1097 or the product will be greater than 9876.This is smaller than the smallest possible multiplicand.

The four digit multiplicand
- cannot end in 1, since then the product will be the same as the single digit multiplicand.
- if multiplied by 7 cannot be greater than 1398
- if multiplied by 6 cannot be greater than 1598
- if multiplied by 4 cannot be greater than 2398
- if multiplied by 3 cannot be greater than 2987
- if multiplied by 2 cannot be greater than 4938

Guess and check is the only way to solve this question but that doesn't mean you can't make an educated guess. Using a spreadsheet would also will help you find your answer quicker.

Good luck finding your solution,

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