如何把 1/7 拆分成四个不同单位分数的和，并且四个分母之和为最小？

ysr

原理和方法仍然有待研究，网友的方法很有效（不过我仍然不太明白），其中的某些参数需要研究和修改。
经过验证1/7拆分成4个单位分数的和，其分母的最小和值是120，但，不懂计算原理，我仍然无法证明！

学习了，谢谢网友！给您点赞！

ysr

这回可以输出最小解了
输入：2
输出：
2  41  {4,10,12,15}
解的个数为： 3组

输入：7
输出：
7  120  {21,24,35,40}
解的个数为： 12组

输入：8
输出：
8  135  {24,30,36,45}
解的个数为： 20组

输入：9
输出：
9  153  {26,36,39,52}
解的个数为： 18组

输入：10
输出：
10  168  {30,40,42,56}
解的个数为： 22组

输入：11
输出：
11  180  {36,44,45,55}
解的个数为： 7组

是参考网友7#楼的代码，虽然不太懂

谢谢网友！！

稍加修改后的代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)

Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(6 * n)
c = Val(b + 1)

Do While c <= Val(10 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

ysr

如何把 1/7 拆分成四个不同单位分数的和，并且四个分母之和在某整数内为最大：
输入：7和50000
输出：
7  24441  {12,29,40,24360}
解中分母的和值不同的解的个数为： 7组

输入：7和20000
输出：
7  18549  {16,20,33,18480}
解中分母的和值不同的解的个数为： 8组

输入：7和90000
输出：
7  24441  {12,29,40,24360}
解中分母的和值不同的解的个数为： 7组

输入：7和210000
输出：
7  24441  {12,29,40,24360}
解中分母的和值不同的解的个数为： 7组

输入：7和310000
输出;
7  24441  {12,29,40,24360}
解中分母的和值不同的解的个数为： 7组

参数不变的情况下是有最大和值解的，求最大和值，参数也要变化

程序代码如下：
Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)
m1 = Val(Text3)
Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(6 * n)
c = Val(b + 1)

Do While c <= Val(10 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k >= m And k <= m1 Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "解中分母的和值不同的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

ysr

输入：7和310000
输出：
7  145055  {9,46,100,144900}
需要比较的解的个数为： 3组

参数修改后的代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)
m1 = Val(Text3)
Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(10 * n)
c = Val(b + 1)

Do While c <= Val(20 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k >= m And k <= m1 Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "需要比较的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

ysr

没有最大和值，即使某数内的最大和值也需要调整参数才能找到，否则照出来的也不一定对

ysr

输入：7和310000
输出：
7  306778  {9,34,429,306306}
需要比较的解的个数为： 3组

稍微改了一下参数，结果就不一样了，这是最大和值吗？不一定，找最大和值，没有意义了！（还是找最小值有意义）

代码如下：
Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)
m1 = Val(Text3)
Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(10 * n)
c = Val(b + 1)

Do While c <= Val(700 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k >= m And k <= m1 Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "需要比较的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

ysr

13  216  {39,52,60,65}
需要比较的解的个数为： 9组
这也是四个分母的最小和值。

代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)
m1 = Val(Text3)
Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(10 * n)
c = Val(b + 1)

Do While c <= Val(50 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And u Mod Abs(v) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "需要比较的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

ysr

输入：100
输出：
100  1611  {350,406,420,435}   不知道是否正确，需要验证一下！
需要比较的解的个数为： 66组

代码如下：

Private Sub Command1_Click()
Dim n, a As Double
n = Val(Text1)
m = Val(24 * n)
a = Val(1 + n)
m1 = Val(Text3)
Do While a >= Val(n) And a <= 5 * n
b = Val(a + 1)


Do While b <= Val(5 * n)
c = Val(b + 1)

Do While c <= Val(10 * n)

u = Val(a * b * c * n)
v = Val(a * b * c - n * (a * b + (a + b) * c))
If Val(v) > 0 Then
d = Val(u / v)

If c < d And Val(u - Int(u / v) * Abs(v)) = 0 Then
k = Val(a + b + c + d)
If k <= m Then
m = k
s = "{" & a & "," & b & "," & c & "," & d & "}"
s2 = n & "  " & m & "  " & s & vbCrLf
s3 = s3 + 1
End If
End If
End If

c = Val(c + 1)
Loop
b = Val(b + 1)
Loop


a = Val(a + 1)
Loop
If s3 > 0 Then
Text2 = s2 & "需要比较的解的个数为： " & s3 & "组"
Else
Text2 = "无解"
End If


End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
End Sub

ysr

ysr 发表于 2026-1-17 08:13
输入：100
输出：
100  1611  {350,406,420,435}   不知道是否正确，需要验证一下！

经过验证100  1611  {350,406,420,435}，这一组解是对的
也就是1/350+1/406+1/420+1/435=8526/852600=1/100

ysr

网友点评：”正确——{100, 1611, {350, 406, 420, 435}——7#代码有瑕疵。“
答：谢谢朋友的指导和鼓励！
7#的代码也不是有瑕疵，仅仅是参数需要调整而已，设a <b< c< d，我认为a≥n+1，a的最低值应该取到n+1，而不是仅仅2*n以上的数，
随着整数的增大，其他参数也可能需要微量调整。