1
vote
2answers
109 views

How many $7$ digits number can be made?

How many $7$ digits number can be made with $1,2,3,4,5,6,7$ so that they are divisible by $11$? (Repetition is not allowed.) I know the divisibility rule of $11$, so the main problem is counting.
2
votes
4answers
150 views

Divisibility by seven

Given number n, whose decimal representation contains digits only $1, 6, 8, 9$. Rearrange the digits in its decimal representation so that the resulting number will be divisible by 7. If number is m ...
1
vote
5answers
140 views

How to show $(mn)!$ divides $(m!)^n$?

How to show $(mn)!$ divides $(m!)^n$, $m$ and $n$ is integers?
3
votes
3answers
243 views

Which Digit-Permutations Preserve Divisibility?

This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it. Let $n > 1,b > 1$ be integers and ...
2
votes
2answers
278 views

Sum of Digits divided by 5 and 9?

Using the digits $0,1,2,3,4,5,6,7,8,9$, If five digit numbers is made without the repetition: How many numbers can be made? sum of all the even numbers? sum of all the odd numbers? How many numbers ...