Let S be a set of all sixdigit integers.
Let S1 be a subset of S, including all members of S such that each consists
of distinct digits.
Let S2 be a subset of S1, including all members of S1 each with 5 being the difference between its largest digit and its lowest one.
Let S3 be a subset of S2, comprising all elements of S2 divisible by 143.
What is the cardinality of S3 ?
Explain your way of reasoning.
Integers of S3 have digits d through (d5) inclusive.
If a member is divisible by 143=11*13 it's divisible by 11.
That means the sum of the differences of pairs of digits will equal a multiple of 11.
The set of digits can be replaced with (0,1,2,3,4,5) without changing divisibility status, and it takes only a couple of minutes handwork to verify the impossibility of divisibility, so S3 contains no members.
Edited on July 6, 2020, 4:01 pm

Posted by xdog
on 20200706 15:53:37 