</pre><pre class="sio" sizcache="27" sizset="64" name="code"><p style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 18px; BORDER-RIGHT-WIDTH: 0px; BACKGROUND-COLOR: rgb(244,195,178); LIST-STYLE-TYPE: none; MARGIN-TOP: 0px; WORD-WRAP: normal; MARGIN-BOTTOM: 5px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; WORD-BREAK: normal; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px"><span style="LINE-HEIGHT: 24px; WORD-WRAP: normal; FONT-SIZE: 14px; WORD-BREAK: normal"><span style="FONT-FAMILY: Microsoft YaHei; COLOR: #006600">快速求正整数次幂,当然不能直接死乘。举个例子:</span></span></p><p style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 18px; BORDER-RIGHT-WIDTH: 0px; BACKGROUND-COLOR: rgb(244,195,178); LIST-STYLE-TYPE: none; MARGIN-TOP: 0px; WORD-WRAP: normal; MARGIN-BOTTOM: 5px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; WORD-BREAK: normal; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px"><span style="LINE-HEIGHT: 24px; WORD-WRAP: normal; FONT-SIZE: 14px; WORD-BREAK: normal"><span style="FONT-FAMILY: Microsoft YaHei; COLOR: #006600">3 ^ 999 = 3 * 3 * 3 * … * 3</span></span></p><p style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 18px; BORDER-RIGHT-WIDTH: 0px; BACKGROUND-COLOR: rgb(244,195,178); LIST-STYLE-TYPE: none; MARGIN-TOP: 0px; WORD-WRAP: normal; MARGIN-BOTTOM: 5px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; WORD-BREAK: normal; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px"><span style="LINE-HEIGHT: 24px; WORD-WRAP: normal; FONT-SIZE: 14px; WORD-BREAK: normal"><span style="FONT-FAMILY: Microsoft YaHei; COLOR: #006600">直接乘要做998次乘法。但事实上可以这样做,先求出2^k次幂:</span></span></p><p style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 18px; BORDER-RIGHT-WIDTH: 0px; BACKGROUND-COLOR: rgb(244,195,178); LIST-STYLE-TYPE: none; MARGIN-TOP: 0px; WORD-WRAP: normal; MARGIN-BOTTOM: 5px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; WORD-BREAK: normal; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px"><span style="LINE-HEIGHT: 24px; WORD-WRAP: normal; FONT-SIZE: 14px; WORD-BREAK: normal"><span style="FONT-FAMILY: Microsoft YaHei; COLOR: #006600">3 ^ 2 = 3 * 3 3 ^ 4 = (3 ^ 2) * (3 ^ 2) 3 ^ 8 = (3 ^ 4) * (3 ^ 4) 3 ^ 16 = (3 ^ 8) * (3 ^ 8) 3 ^ 32 = (3 ^ 16) * (3 ^ 16) 3 ^ 64 = (3 ^ 32) * (3 ^ 32) 3 ^ 128 = (3 ^ 64) * (3 ^ 64) 3 ^ 256 = (3 ^ 128) * (3 ^ 128) 3 ^ 512 = (3 ^ 256) * (3 ^ 256)</span></span></p><p style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 18px; BORDER-RIGHT-WIDTH: 0px; BACKGROUND-COLOR: rgb(244,195,178); LIST-STYLE-TYPE: none; MARGIN-TOP: 0px; WORD-WRAP: normal; MARGIN-BOTTOM: 5px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; WORD-BREAK: normal; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px"><span style="LINE-HEIGHT: 24px; WORD-WRAP: normal; FONT-SIZE: 14px; WORD-BREAK: normal"><span style="FONT-FAMILY: Microsoft YaHei; COLOR: #006600">再相乘:</span></span></p><p style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 18px; BORDER-RIGHT-WIDTH: 0px; BACKGROUND-COLOR: rgb(244,195,178); LIST-STYLE-TYPE: none; MARGIN-TOP: 0px; WORD-WRAP: normal; MARGIN-BOTTOM: 5px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; WORD-BREAK: normal; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px"><span style="LINE-HEIGHT: 24px; WORD-WRAP: normal; FONT-SIZE: 14px; WORD-BREAK: normal"><span style="FONT-FAMILY: Microsoft YaHei; COLOR: #006600">3 ^ 999 = 3 ^ (512 + 256 + 128 + 64 + 32 + 4 + 2 + 1) = (3 ^ 512) * (3 ^ 256) * (3 ^ 128) * (3 ^ 64) * (3 ^ 32) * (3 ^ 4) * (3 ^ 2) * 3</span></span></p><p style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 18px; BORDER-RIGHT-WIDTH: 0px; BACKGROUND-COLOR: rgb(244,195,178); LIST-STYLE-TYPE: none; MARGIN-TOP: 0px; WORD-WRAP: normal; MARGIN-BOTTOM: 5px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; WORD-BREAK: normal; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px"><span style="LINE-HEIGHT: 24px; WORD-WRAP: normal; FONT-SIZE: 14px; WORD-BREAK: normal"><span style="FONT-FAMILY: Microsoft YaHei; COLOR: #006600">这样只要做16次乘法。即使加上一些辅助的存储和运算,也比直接乘高效得多(尤其如果这里底数是成百上千位的大数字的话)。</span></span></p><p style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 18px; BORDER-RIGHT-WIDTH: 0px; BACKGROUND-COLOR: rgb(244,195,178); LIST-STYLE-TYPE: none; MARGIN-TOP: 0px; WORD-WRAP: normal; MARGIN-BOTTOM: 5px; BORDER-TOP-WIDTH: 0px; BORDER-BOTTOM-WIDTH: 0px; WORD-BREAK: normal; BORDER-LEFT-WIDTH: 0px; PADDING-TOP: 0px"><span style="LINE-HEIGHT: 24px; WORD-WRAP: normal; FONT-SIZE: 14px; WORD-BREAK: normal"><span style="FONT-FAMILY: Microsoft YaHei; COLOR: #006600">我们发现,把999转为2进制数:1111100111,其各位就是要乘的数。这提示我们利用求二进制位的算法(其中mod是模运算):</span></span></p>
#include <stdio.h>
#include <string.h>
typedef __int64 int64; //不能用#define
int64 mult(int64 a,int64 b,int64 m)
{
int64 ans=1;
while(b)
{
if(b%2==1) //b%2==1 与 b&1 等价,所以也可以用 b&1
{
ans=(ans*a)%m;
}
b=b/2; //b=b/2 与 b>>=1 等价。
a=(a*a)%m;
}
return ans%m;
}
int main()
{
int i,z,m,h,a[5000],b[5000];
int64 sum;
scanf("%d",&z);
while(z--)
{
sum=0;
scanf("%d%d",&m,&h);
for(i=0;i<h;i++)
scanf("%d%d",&a[i],&b[i]);
for(i=0;i<h;i++)
sum=(sum+mult(a[i],b[i],m))%m;
printf("%I64d\n",sum);
}
return 0;
}