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 Two-three integers (Posted on 2017-10-15)
Call an integer a two-three integer if it is greater than 1 and has the form 2a * 3b, where a ≥ 0 and b ≥ 0.
Express 19992000 as a sum of two-three integers so that none of the addends divides another.

Source: Eighth Korean Mathematical Olympiad (Crux Math. Dec 1999)

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer-aided solution | Comment 1 of 6
The theory behind the solution:

If the two-three integers are arranged with decreasing powers of 3, they must necessarily have increasing powers of 2 so that with any pair of numbers neither will divide the other, as one will have a higher power of 3 while the other has a higher power of 2.

The program below lists all such pairings of powers where the resulting integers add to 19992000 (I assume in honor of the pseudo millennium change).

There are 31 solutions.  Each row in the table shows the power of two followed by the power of three in each of the component two-three integers.  For example, the first one is:

`2^6*3^11 + 2^8*3^9 + 2^10*3^7 + 2^16*3^2 + 2^18*3^1  or11337408 + 5038848 + 2239488 +  589824  +  786432  = 19992000`

6 11    8 9    10 7    16 2    18 1
6 11    8 9    10 5    13 3    20 1
6 11    8 8    9 7    11 6    12 5    13 3    20 1
6 11    8 8    9 7    11 6    12 4    15 3    20 1
6 11    8 8    9 7    11 5    14 4    15 3    20 1
6 11    8 8    9 5    11 4    17 3    20 1
6 11    8 7    10 6    13 5    14 4    15 3    20 1
6 11    8 7    10 6    13 4    17 3    20 1
6 10    7 9    9 8    12 7    16 2    18 1
6 10    7 9    9 8    12 6    13 5    14 4    15 3    20 1
6 10    7 9    9 8    12 6    13 4    17 3    20 1
6 10    7 9    9 8    12 5    15 4    17 3    20 1
6 10    7 9    9 6    14 5    15 4    17 3    20 1
6 10    7 9    9 6    14 3    15 2    22 1
6 10    7 8    8 7    11 6    14 5    15 4    17 3    20 1
6 10    7 8    8 7    11 6    14 3    15 2    22 1
6 10    7 8    8 7    11 5    12 4    13 3    17 2    22 1
6 10    7 8    8 7    11 4    15 3    17 2    22 1
6 10    7 8    8 5    13 4    15 3    17 2    22 1
6 10    7 7    9 6    10 5    13 4    15 3    17 2    22 1
6 9    8 8    10 7    11 6    14 5    15 4    17 3    20 1
6 9    8 8    10 7    11 6    14 3    15 2    22 1
6 9    8 8    10 7    11 5    12 4    13 3    17 2    22 1
6 9    8 8    10 7    11 4    15 3    17 2    22 1
6 9    8 8    10 6    12 5    13 4    15 3    17 2    22 1
6 9    8 7    9 6    11 5    19 3    20 1
6 9    8 7    9 6    11 5    19 2    22 1
6 8    7 7    13 6    14 5    15 4    17 3    20 1
6 8    7 7    13 6    14 3    15 2    22 1
6 8    7 7    13 5    19 3    20 1
6 8    7 7    13 5    19 2    22 1

DefDbl A-Z
Dim crlf\$, pwrs(100, 2 To 3), totSoFar, goal

Form1.Visible = True

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

goal = 19992000

pwrs(0, 2) = -1
pwrs(0, 3) = 16

Text1.Text = Text1.Text & crlf & " done"

End Sub

If totSoFar > goal Then Exit Sub

For b = pwrs(wh - 1, 3) - 1 To 0 Step -1
For a = pwrs(wh - 1, 2) + 1 To 24
n = Int(2 ^ a * 3 ^ b + 0.5)
totSoFar = totSoFar + n
If n > goal Then totSoFar = totSoFar - n: Exit For
pwrs(wh, 2) = a: pwrs(wh, 3) = b
If totSoFar = goal Then
For i = 1 To wh
Text1.Text = Text1.Text & pwrs(i, 2) & Str(pwrs(i, 3)) & "    "
Next
Text1.Text = Text1.Text & crlf
totSoFar = totSoFar - n: Exit For
End If
DoEvents
totSoFar = totSoFar - n
Next
Next
End Sub

Edited on October 15, 2017, 4:55 pm
 Posted by Charlie on 2017-10-15 16:54:45

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