已知 m m m, n n n为正整数，且 m 2 + n 2 = 2009 m^2+n^2=2009 m2+n2=2009,求 m + n m+n m+n的值。
一、使用因式分解加枚举
① 对2009进行短除
7 | 2009 ‾ 7 | 287 ‾ 41 7 \underbar{|\enspace\enspace2009\enspace} \\ \enspace\enspace 7 \underbar{|\enspace287\enspace}\\ \enspace\enspace\enspace\enspace\enspace 41 \enspace 7|20097|28741
② 整理
2009
=
7
2
×
(
a
2
+
b
2
)
=
7
2
×
a
2
+
7
2
×
b
2
=
(
7
a
)
2
+
(
7
b
)
2
2009=7^2×(a^2+b^2)=7^2×a^2+7^2×b^2=(7a)^2 +(7b)^2
2009=72×(a2+b2)=72×a2+72×b2=(7a)2+(7b)2
则
m
=
7
a
m=7a
m=7a,
n
=
7
b
n=7b
n=7b,
a
2
+
b
2
=
41
a^2+b^2=41
a2+b2=41
③ 根据题目限定范围为正整数进行枚举
当
a
=
1
a=1
a=1,
b
=
±
40
b=±\sqrt{40}
b=±40 (不成立)
当
a
=
2
a=2
a=2,
b
=
±
37
b=±\sqrt{37}
b=±37 (不成立)
当
a
=
3
a=3
a=3,
b
=
±
32
b=±\sqrt{32}
b=±32 (不成立)
当
a
=
4
a=4
a=4,
b
=
±
25
=
±
5
b=±\sqrt{25}=±5
b=±25=±5 (成立)
当
a
=
5
a=5
a=5,
b
=
±
16
=
±
4
b=±\sqrt{16}=±4
b=±16=±4 (成立)
当
a
=
6
a=6
a=6,
b
=
±
5
b=±\sqrt{5}
b=±5 (不成立)
当
a
=
7
a=7
a=7,
b
=
±
−
8
b=±\sqrt{-8}
b=±−8 (不成立)
④ 根据枚举结果和题目限定范围为正整数
当
a
=
4
a=4
a=4,
b
=
5
b=5
b=5 则
m
=
28
m=28
m=28,
n
=
35
n=35
n=35 即
m
+
n
=
28
+
35
=
63
m+n=28+35=63
m+n=28+35=63
当
a
=
5
a=5
a=5,
b
=
4
b=4
b=4 则
m
=
35
m=35
m=35,
n
=
28
n=28
n=28 即
m
+
n
=
38
+
28
=
63
m+n=38+28=63
m+n=38+28=63
所以：
m
+
n
=
63
m+n=63
m+n=63
二、使用编程进行枚举
public static void Calc()
{
int max = (int)Math.Ceiling(Math.Sqrt(2009));
for (int n = 0; n < max; n++)
{
for (int m = 0; m < max; m++)
{
double v = Math.Pow(n,2) + Math.Pow(m, 2);
if(v== 2009)
{
System.Diagnostics.Debug.WriteLine($"n+m={n+m}");
}
}
}
}
运行结果
n+m=63
n+m=63
三、使用图形观察
圆与第一象限中整数相交的点为答案，将点的 x x x与 y y y进行相加得到计算结果
using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Drawing;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows.Forms;
namespace Subject3LightSimulation
{
public partial class Form3 : Form
{
float radius = 0;
public Form3()
{
InitializeComponent();
radius =(float) Math.Sqrt(2009);
}
protected override void OnPaint(PaintEventArgs e)
{
base.OnPaint(e);
using (Brush b = new SolidBrush(Color.FromArgb(192,192,192,192)))
{
using (Pen p = new Pen(b))
{
p.DashStyle = System.Drawing.Drawing2D.DashStyle.DashDot;
for (int i = 0; i < 100; i++)
{
e.Graphics.DrawLine(p, new Point(0, i * 10), new Point(1000, i * 10));
}
for (int i = 0; i < 100; i++)
{
e.Graphics.DrawLine(p, new Point(i * 10, 0), new Point(i * 10, 1000));
}
}
}
using (Brush b = new SolidBrush(Color.Blue))
{
using (Pen p = new Pen(b))
{
e.Graphics.DrawLine(p, new Point(0, 500), new Point(1000,500));
e.Graphics.DrawLine(p, new Point(500, 0), new Point(500, 1000));
}
}
using (Brush b2 = new SolidBrush(Color.Red))
{
using (Pen p2 = new Pen(b2))
{
float x = 500 - radius*10;
float y = 500 - radius * 10;
float w = 2 * radius * 10;
float h = 2 * radius * 10;
float s = 0.0f;
float e2 = 360;
e.Graphics.DrawArc(p2, x,y,w,h,s,e2);
}
}
using (Brush b2 = new SolidBrush(Color.Red))
{
using (Pen p2 = new Pen(b2))
{
float x = 500 - 1 * 10;
float y = 500 - 1 * 10;
float w = 2 * 1 * 10;
float h = 2 * 1 * 10;
float s = 0.0f;
float e2 = 360;
e.Graphics.DrawArc(p2, x, y, w, h, s, e2);
}
}
using (Brush b2 = new SolidBrush(Color.Green))
{
using (Pen p2 = new Pen(b2))
{
int x = 500 + 28 * 10;
int y = 500 - 35 * 10;
e.Graphics.DrawLine(p2, new Point(500, 500), new Point(x, y));
int x2 = 500 + 35 * 10;
int y2 = 500 - 28 * 10;
e.Graphics.DrawLine(p2, new Point(500, 500), new Point(x2, y2));
}
using (Pen p3 = new Pen(b2))
{
p3.DashStyle = System.Drawing.Drawing2D.DashStyle.DashDot;
int x = 500 + 28 * 10;
int y = 500 - 35 * 10;
e.Graphics.DrawLine(p3, new Point(x, 500), new Point(x, y));
e.Graphics.DrawLine(p3, new Point(x, y), new Point(500, y));
int x2 = 500 + 35 * 10;
int y2 = 500 - 28 * 10;
e.Graphics.DrawLine(p3, new Point(x2, 500), new Point(x2, y2));
e.Graphics.DrawLine(p3, new Point(x2, y2), new Point(500, y2));
}
}
using (Brush b2 = new SolidBrush(Color.Green))
{
using (Pen p2 = new Pen(b2))
{
int x = 500 + 28 * 10;
int y = 500 - 35 * 10;
e.Graphics.DrawString("28",Font, b2, new Point(x, 500));
e.Graphics.DrawString("35", Font, b2, new Point(500, y));
e.Graphics.DrawString("_____", Font, b2, new Point(x/2 + 250+ (int)Font.Size + 1, y/2 + 250-Font.Height+3));
e.Graphics.DrawString("√2009", Font, b2, new Point(x/2 + 250, y/2 + 250));
int x2 = 500 + 35 * 10;
int y2 = 500 - 28 * 10;
e.Graphics.DrawString("35", Font, b2, new Point(x2, 500));
e.Graphics.DrawString("28", Font, b2, new Point(500, y2));
e.Graphics.DrawString("_____", Font, b2, new Point(x2/2 + 250 + (int)Font.Size+1, y2/2 + 250 - Font.Height + 3));
e.Graphics.DrawString("√2009", Font, b2, new Point(x2/2+250, y2/2 + 250));
}
}
}
}
}
运行结果
扩展
在html中使用table元素表示短除
<html>
<body>
<table>
<tr>
<td width="40px"></td>
<td width="20px"></td>
<td width="40px"></td>
</tr>
<tr>
<td style="text-align:right">7</td>
<td style="text-algin:center;border-left:2px solid #000;border-bottom:2px solid #000;" align="center" colspan="3">2009</td>
</tr>
<tr>
<td></td>
<td style="text-align:right">7</td>
<td style="text-algin:center;border-left:2px solid #000;border-bottom:2px solid #000;" align="center" colspan="2">287</td>
</tr>
<tr>
<td ></td>
<td ></td>
<td style="text-align:center">41</td>
</tr>
</table>
</body>
</html>
使用上述代码亲自试一试
显示效果
