# Question on Number theory

1. Jan 30, 2013

### hms.tech

1. The problem statement, all variables and given/known data

How many four-digit numbers formed of only odd digits are divisible by five?

2. Relevant equations

Permutations

3. The attempt at a solution

Here is what I think should be done :

Ans : 4P3 * 1
= 24

Is that right ?

2. Jan 30, 2013

### HallsofIvy

Staff Emeritus
You don't say how you got that number so I don't see any way to comment except to say that 4P3= 4 is clearly NOT the correct answer. You don't say why you think that is true. How did you get that?

There are a total of 4 digits in the number and 5 odd digits. How many choices are there for the first digit? The second ? The third? The fourth?

3. Jan 30, 2013

### tiny-tim

why?

4. Jan 30, 2013

### hms.tech

here is how I did it :

The last digit is reserved for "5" since we want it to be divisible by "5"

Then, the choices for the first digit are : 4
2nd digit : 3
3rd digit : 2
Ergo, 4P3 * 1 = 4P3 = 24

I am honestly surprised why this method is incorrect .

5. Jan 30, 2013

### tiny-tim

ah, you're assuming they all have to be different

they don't

(btw, i'm not familiar with this 4P3 notation, but it doesn't look right …

24 = 4!, so where does 3 come into it? )

6. Jan 30, 2013

### hms.tech

Hmmm...I think you are right, I must not assume this.

Honestly, I have no experience how to calculate the permutations if digits can be repeated.

I'll try anyway :

There are 5 choices for each of the first three digits and one choice for the last digit.
5*5*5*1 = 125

7. Jan 30, 2013

### Dick

Right!

8. Jan 30, 2013

### tiny-tim

Woohoo!

9. Jan 31, 2013

### HallsofIvy

Staff Emeritus
If the digits can be repeated, this is NOT a permutations problem. It is simply a application of the "fundamental counting principle": if A can be done in m ways and B can be done, independently, in n ways the A and B can be done in mn ways.
There are 5 ways to choose the first digit, 5 ways to choose the second digit, 5 ways to choose the third digit, and only one way to choose the last digit which u must be 5.
5(5)(5)(1)=

Exactly right.