novaBBS - rec.puzzles - (natural number) X has 4 factors, which add up to 400

(natural number) X has 4 factors, which add up to 400

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Subject: (natural number) X has 4 factors, which add up to 400
From: henha...@gmail.com (henh...@gmail.com)
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by: henh...@gmail.com - Thu, 4 Aug 2022 16:18 UTC

Find all positive integers X such that...

X has 4 factors, and

the 4 factors add up to 400

( 6 has 4 factors: 1, 2, 3, 6)

------- pls wait a few days before posting answers or hints.

Re: (natural number) X has 4 factors, which add up to 400

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Subject: Re: (natural number) X has 4 factors, which add up to 400
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From: news.dea...@darjeeling.plus.com (Mike Terry)
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by: Mike Terry - Thu, 4 Aug 2022 19:44 UTC

On 04/08/2022 17:18, henh...@gmail.com wrote:
>
> Find all positive integers X such that...
>
> X has 4 factors, and
>
> the 4 factors add up to 400
>
>
>
> ( 6 has 4 factors: 1, 2, 3, 6)
>
>
> ------- pls wait a few days before posting answers or hints.
>

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Repeating problem statement:

> Find all positive integers X such that...
>
> X has 4 factors, and
>
> the 4 factors add up to 400
>
>
>
> ( 6 has 4 factors: 1, 2, 3, 6)

For exactly 4 factors, X will need to have either 1 or 2 prime factors, making X have one of the
following forms (where p,q are distinct primes):

A) X = pq
factors: 1,p,q,pq

B) X = p^3
factors: 1,p,p^2,p^3 [I nearly overlooked this possibility!]

For (A), 400 = 1+p+q+pq = (1+p)(1+q), and testing p up to 20 there are clearly no solutions. E.g.
p=3 passes first test: 1+p = 4 | 400, but that would make q=99 which is not prime. Nearly working
is p=19, but that would make q=19, which is prime, but p and q need to be distinct factors of X.

For (B) trial and error shows p must be around 7, and in fact p=7 works!! 1 + 7 + 49 + 343 = 400.

So... only solution is X = 343.

Mike.

"I'm not a god, I was misquoted." -- Lister, Red Dwarf

interests / rec.puzzles / (natural number) X has 4 factors, which add up to 400

Subject	Author
(natural number) X has 4 factors, which add up to 400	henh...@gmail.com
Re: (natural number) X has 4 factors, which add up to 400	Mike Terry