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Powerful and consecutive (Posted on 2010-08-09) Difficulty: 4 of 5
Powerful numbers (4,8,9,16,25,27... )are defined as follows: if a prime p divides n then p2 must also divide n.
(8,9) is a couple of two consecutive numbers,both of them being powerful.
Find another pair(s) like that.

The more the merrier!!

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (2 votes)

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Go by definition | Comment 4 of 16 |
According to the definition, 36 is a powerful number. (2 and 3 are the only primes which divide it, and 2^2 and 3^2 divide as well.) I think it's just because the first few powerful numbers are perfect powers, that these weren't included in the examples.

However note numbers like n=20 are not powerful by that definition, as 5 is a prime which divides n, but p^2 does not divide n. The definition "If a prime factor does this, it must also do this" is actually a statement over all prime factors. However of course, most (almost all) of them do not divide n, and so do not apply.

  Posted by Gamer on 2010-08-09 13:39:36
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