# Number Theory

1. Feb 7, 2005

1. Find seven different unit fractions whose sum is 1. So

$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} + \frac{1}{f} + \frac{1}{g} = 1$$ Would this just be purely guess and check?

2. How many different 6 digit numbers can you make using $$1,2,5,6,7,9$$. Would it just be $$6!$$? Also how would you find how many of these numbers are divisble by 6?

Thanks!

2. Feb 7, 2005

### vincentchan

yes, the combination is 6!

all 6 digits numbers form by those numbers are divisible by 3 (because 1,2,5,6,7,9 add up to 30, which is divisible by 3) therefore, all 6 digits numbers which ended with 2 and 6 are divisible by 6

3. Feb 7, 2005

but then for #2 how do you find how many are divisble by 6

also for #1 I dont see another way than gues and check.

thanks

4. Feb 7, 2005

### Curious3141

For the second part, what are the number of possibilities with a terminal digit of 2 ? With a terminal digit of 6 ? Add these up.

It should be 2*(5!)

5. Feb 7, 2005

Thanks a lot

So I guess for the first one we have to guess and check. Just take fractions see if they add to 1, and then take the reciprocal of the reciprocal?

6. Feb 8, 2005

### Galileo

The question doesn't state explicitly that each number may only be used once.
So then 111111 and 965759 etc. are also solutions and the answer would be $$6^6$$.

7. Feb 8, 2005

thanks a lot. i guess the second question is impossible then?

8. Feb 8, 2005