Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 230

how many positive integers

For how many positive integers n is

an integer ?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: how many positive integers

If n is a positive even integer,

, which is not divisible by

.

Now suppose n is a positive odd integer.

When

.

, which is divisible by 2, 8 and 32, but not by 128.

Suppose

is divisible by by 2, 8 and 32 but not by 128 for some odd integer n.

Then

is also divisible by 2, 8 and 32 but not by 128.

Hence, by induction,

is divisible by 2, 8 and 32 but not by 128 for all positive odd integers n.

So there are only three positive integers n (namely

) for which

is an integer.

tony123

Member

Registered: 2007-08-03

Posts: 230

Re: how many positive integers

Very nice work Jane thank you