Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number n greater than 10. Then, she multiplies all the digits of n and obtains an odd number. Find all possible values of the units digit of n.
It looks like 5 is the only possible answer.
First off, 5 is a possible answer if the original numbers were 3 and 5.
n=15 and the product 1*5 is odd.
If original set contains a 5, the product will end in 5 or 0. But it can't end in 0 or the product of the digits will be even. Therefore n will end in 5.
If the original set contains an even digit, n will be even and the product of n's digits will be even.
So we only need to consider the original set consisting of {1,3,7,9}
But 1's won't affect the product and 9=3x3.
The set further reduces to {3,7}
There are three possibilities
1) No product of 3^a*7^b contains all odd digits.
2) Any product containing all odd digits contains a 5.
3) There is a product containing all odd digits that does not include a 5.
A quick scan of powers up to about 15 shows all numbers with at least one even digit. As these numbers increase, finding one with odd digits becomes less and less likely. So it looks like possibility 1) is the case.
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Posted by Jer
on 2024-05-24 09:43:00 |
Conjecture
