good one

A positive integer n is a good number if it can be written as the sum of two positive integers a and b, i.e. n=a+b, such that n|ab. It is a very good number if a,b are distinct. Find all numbers n such that n is good but it's not very good.

7 Answers

1
Rohan Ghosh ·

Doesnt it reduce to the already known equation ??

(a+b)|ab

=>

(a+b)*k=ab

1/a+1/b=1/k {Already known eq.}

further if it is also very good (which is not desired)

n=2a
2a|a^2

=>
2|a

thus n=2*2k=4k

thus numbers divisible by four are excluded in the list :)

39
Dr.House ·

We must now find all numbers of the form 4k which are not very good.

take a careful look.

1
Rohan Ghosh ·

you see any number of the form 4k is very good !!
4k=2k+2k

and

4k|2k*2k

as 2k*2k/4k=k :P

9
Celestine preetham ·

n = a2/λ where a is odd , λ<a

the proof is interesting

juniors this is solvable unlike other very tough olympiad Qs

pls try ....

39
Dr.House ·

yes please try

11
Devil ·

Just a minute, I cud not really get it...
n has to be good...meanng n|ab, so ab=nk.... and n=a+b=2a (Since n is not very good).
So i need a^2/2a=a/2 as an integer which is true for infinitely many such n's?????

9
Celestine preetham ·

yeah i think rohan and i have both taken the opposite !

Your Answer

Close [X]