Exploring Number Theory: Finding Unit Fractions and Divisible Numbers

[courtrigrad]

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!

 

[vincentchan]

i think i can answer your 2nd question

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

 

[courtrigrad]

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

also for #1 I don't see another way than gues and check.

thanks

 

[Curious3141]

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

also for #1 I don't see another way than gues and check.

thanks

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!)

 

[courtrigrad]

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?

 

[Galileo]

2. How many different 6 digit numbers can you make using $$1,2,5,6,7,9$$. Would it just be $$6!$$?

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 $$6^6$$.

 

[courtrigrad]

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

 

[courtrigrad]

I think you have to simplify all of he fractions for #2. IS this right?

Thanks