how many of the resulting number are divisible by 11 but not by 5?

Question

To me, this is a hard problem. Can anyone help me with this?

The number 3456 is divisible by 11 and by 5. Give that you change the position of two or more of these four digits, how many of the resulting number are divisible by 11 but not by 5?

Answer 1

First of all, the number $3456$ is not divisible by 11 and 5.

But if we change position of digits then the numbers divisible by $11$ are-

$3564÷11=324$

$3465÷11=315\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;3465÷5=693$

$4356÷11=396$

$4653÷11=423$

$5346÷11=486$

$5643÷11=513$

$6534÷11=594$

$6435÷11=585\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;6435÷5=1287$

But the numbers $3465$ and $6435$ are also divisible by $5$. So the numbers divisible by only $11$ are-

$3564,\;4356,\;4653,\;5346,\;5643,\;and\;6534$

I hope it' ll help you.

Answer 2

A number is divisible by 5 if and only if it ends in 0 or 5. A number is divisible by 11 if and only if the alternating sum of its digits is 0.

Knowing that there are only 24 possible rearrangements, you can use these facts to find all such numbers.