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## #1 2008-11-08 23:01:05

tony123
Member
Registered: 2007-08-03
Posts: 189

### how many positive integers

For how many positive integers n is

an integer ?

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## #2 2008-11-09 04:19:31

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.

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## #3 2008-11-09 07:27:48

tony123
Member
Registered: 2007-08-03
Posts: 189

### Re: how many positive integers

Very nice work Jane thank you

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