Problem:
Find the smallest positive integer whose cube ends in .
Solution:
If the cube of an integer ends in , then the integer itself must end in ; i.e., must be of the form . Therefore,
where the penultimate term, , determines the penultimate digit of , which must also be . In view of this, must also end in ; i.e., must end in or , and hence be of the form . Thus
Since the first two terms on the right of end in , while the last term ends in , it follows that must end in . The smallest which will ensure this is , implying that , and . (Indeed, .)
The problems on this page are the property of the MAA's American Mathematics Competitions