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Topic: Word Problem help
Replies: 7   Last Post: Feb 9, 2012 11:52 AM

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Nat Silver

Posts: 2,082
Registered: 12/6/04
Re: Word Problem help
Posted: Sep 23, 2003 1:18 AM
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It all boils down to whether a given
number has an odd or even number
of distinct factors.

Consider, for example, the number 24.
Pair off its factors, x and y, where xy = 24,
one always is larger than the other:

1, 24
2, 12
3, 8
4, 6

Since 24 has an even number
of factors, the person in seat
24 will return to the original
seated postion.

Most numbers have all their distinct factors
pairing off in this way, larger and smaller.
The numbers which don't are perfect squares.
Consider, for example, 36.

1 36
2 18
3 12
4 9
6 6

Note the repeating 6, the principal square root of 36,
which is the essence of being a perfect square.
Numbers with an odd number of distinct factors are
perfect squares. Those will be left standing.

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