尤拉計劃第四題:關於求迴文數。第五題,第六題,第七題,第八題,第九題,第十題
第四題:迴文數問題
A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 99.
Find the largest palindrome made from the product of two 3-digit numbers.
一個迴文數指的是從左向右和從右向左讀都一樣的數字。最大的由兩個兩位數乘積構成的迴文數是9009 = 91 * 99.
找出最大的有由個三位數乘積構成的迴文數。
這個題目難度不大,寫一個小程式就OK了,
思路分為兩步:
第一:因為999*999=998001,所以從998001開始向前,找回文數,因為我們要的是最大的,所以從大往小找
第二:找到以後驗證是否是兩個三位數的乘積,這裡的驗證也是從大往小,看看能不能被999整除,如果能,檢查商是不是三位數,如果是,那麼結果就出來了。
一下為程式:
use strict; use warnings; my $i; my $flag; my $num; for($i=998001;$i>100000;$i--) { $num=$i; my $six=(int($num/100000)); my $five=(int($num/10000)-10*int($num/100000)); my $four=(int($num/1000)-10*int($num/10000)); my $three=(int($num/100)-10*int($num/1000)); my $two=(int($num/10)-10*int($num/100)); my $one=($num-10*int($num/10)); if($six==$one && $five==$two && $four==$three) { my $cout=999; for($cout=999;$cout>100;$cout--) { if(0==$num%$cout) { my $another=$num/$cout; if($another>100 && $another<999) { print "$num = $cout X $another \n"; $flag=1; last; } } } last if $flag==1; } }
結果如下:
C:\WINDOWS\system32\cmd.exe /c perl "F:\perl\c.pl
906609 = 993 X 913
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第五題:整除問題
題目如下:
2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?
2520是最小的能被1-10中每個數字整除的正整數。
最小的能被1-20中每個數整除的正整數是多少?
這裡的第二題其實都不用謝程式的,直接拿個計算器算一下就好了,
如下:
$i=2*3*2*5*7*2*3*11*13*2*17*19;
print $i."\n";
結果如下:
C:\WINDOWS\system32\cmd.exe /c perl "F:\perl\c.pl"
232792560
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第六題:平方和,和的平方
題目如下:
The sum of the squares of the first ten natural numbers is,
12 + 22 + ... + 102 = 385(平方和)The square of the sum of the first ten natural numbers is,
(1 + 2 + ... + 10)2 = 552 = 3025(和的平方)Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 385 = 2640.(和的平方-平方的和=?)
Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.
求1到100 和的平方-平方的和=?
程式如下:
use strict;
use warnings;
my $sum=0;
my $sum_square=0;
my $square_sum=0;
my $cout;
my $difference;
for($cout=1;$cout<101;$cout++)
{
$square_sum=$square_sum+$cout*$cout;
$sum=$sum+$cout;
}
$sum_square=$sum*$sum;
$difference=$square_sum-$sum_square;
print "$difference = $square_sum-$sum_square\n";
結果如下:
C:\WINDOWS\system32\cmd.exe /c perl "F:\perl
-25164150 = 338350-25502500
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第七題:第10001個素數
題目如下:
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
What is the 10 001st prime number?
也就是著第10001個素數:程式如下:
use strict;
use warnings;
my $cout=1;
my $flag=0;
my $i;
for($i=3;;$i+=2)#這裡是間隔2,這樣可以節省一半的時間
{
foreach(2..$i**0.5)
{
if(0==$i%$_)
{
$flag=1;
last;
}
else
{
next;
}
}
$cout=$cout+1 if($flag==0);
$flag=0;
if($cout==10001)
{
print "$i\n";
last;
}
}
結果如下:
C:\WINDOWS\system32\cmd.exe /c perl "F:\perl\c.pl"
104743
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第八題:找出以下這個1000位的整數中連續5個數字的最大乘積
這個題目非常的長:
如下:
找出以下這個1000位的整數中連續5個數字的最大乘積。(例如前五個數字的乘積是7*3*1*6*7=882)
73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450
我們把所有的數放在一個數組裡,然後從第一個元素開始依次相乘,並將得到的結果進行比較,大的留下。
這裡要說的把1000個數中間都加一個空格,然後變成陣列,用vim的巨集就很簡單的完成了。
程式如下:
my $big=0;
my $sum=0;
my @num=qw/7 3 1 6 7 1 7 6 5 3 1 3 3 0 6 2 4 9 1 9 2 2 5 1 1 9 6 7 4 4 2 6 5 7 4 7 4 2 3 5 5 3 4 9 1 9 4 9 3 4 9 6 9 8 3 5 2 0 3 1 2 7 7 4 5 0 6 3 2 6 2 3 9 5 7 8 3 1 8 0 1 6 9 8 4 8 0 1 8 6 9 4 7 8 8 5 1 8 4 3 8 5 8 6 1 5 6 0 7 8 9 1 1 2 9 4 9 4 9 5 4 5 9 5 0 1 7 3 7 9 5 8 3 3 1 9 5 2 8 5 3 2 0 8 8 0 5 5 1 1 1 2 5 4 0 6 9 8 7 4 7 1 5 8 5 2 3 8 6 3 0 5 0 7 1 5 6 9 3 2 9 0 9 6 3 2 9 5 2 2 7 4 4 3 0 4 3 5 5 7 6 6 8 9 6 6 4 8 9 5 0 4 4 5 2 4 4 5 2 3 1 6 1 7 3 1 8 5 6 4 0 3 0 9 8 7 1 1 1 2 1 7 2 2 3 8 3 1 1 3 6 2 2 2 9 8 9 3 4 2 3 3 8 0 3 0 8 1 3 5 3 3 6 2 7 6 6 1 4 2 8 2 8 0 6 4 4 4 4 8 6 6 4 5 2 3 8 7 4 9 3 0 3 5 8 9 0 7 2 9 6 2 9 0 4 9 1 5 6 0 4 4 0 7 7 2 3 9 0 7 1 3 8 1 0 5 1 5 8 5 9 3 0 7 9 6 0 8 6 6 7 0 1 7 2 4 2 7 1 2 1 8 8 3 9 9 8 7 9 7 9 0 8 7 9 2 2 7 4 9 2 1 9 0 1 6 9 9 7 2 0 8 8 8 0 9 3 7 7 6 6 5 7 2 7 3 3 3 0 0 1 0 5 3 3 6 7 8 8 1 2 2 0 2 3 5 4 2 1 8 0 9 7 5 1 2 5 4 5 4 0 5 9 4 7 5 2 2 4 3 5 2 5 8 4 9 0 7 7 1 1 6 7 0 5 5 6 0 1 3 6 0 4 8 3 9 5 8 6 4 4 6 7 0 6 3 2 4 4 1 5 7 2 2 1 5 5 3 9 7 5 3 6 9 7 8 1 7 9 7 7 8 4 6 1 7 4 0 6 4 9 5 5 1 4 9 2 9 0 8 6 2 5 6 9 3 2 1 9 7 8 4 6 8 6 2 2 4 8 2 8 3 9 7 2 2 4 1 3 7 5 6 5 7 0 5 6 0 5 7 4 9 0 2 6 1 4 0 7 9 7 2 9 6 8 6 5 2 4 1 4 5 3 5 1 0 0 4 7 4 8 2 1 6 6 3 7 0 4 8 4 4 0 3 1 9 9 8 9 0 0 0 8 8 9 5 2 4 3 4 5 0 6 5 8 5 4 1 2 2 7 5 8 8 6 6 6 8 8 1 1 6 4 2 7 1 7 1 4 7 9 9 2 4 4 4 2 9 2 8 2 3 0 8 6 3 4 6 5 6 7 4 8 1 3 9 1 9 1 2 3 1 6 2 8 2 4 5 8 6 1 7 8 6 6 4 5 8 3 5 9 1 2 4 5 6 6 5 2 9 4 7 6 5 4 5 6 8 2 8 4 8 9 1 2 8 8 3 1 4 2 6 0 7 6 9 0 0 4 2 2 4 2 1 9 0 2 2 6 7 1 0 5 5 6 2 6 3 2 1 1 1 1 1 0 9 3 7 0 5 4 4 2 1 7 5 0 6 9 4 1 6 5 8 9 6 0 4 0 8 0 7 1 9 8 4 0 3 8 5 0 9 6 2 4 5 5 4 4 4 3 6 2 9 8 1 2 3 0 9 8 7 8 7 9 9 2 7 2 4 4 2 8 4 9 0 9 1 8 8 8 4 5 8 0 1 5 6 1 6 6 0 9 7 9 1 9 1 3 3 8 7 5 4 9 9 2 0 0 5 2 4 0 6 3 6 8 9 9 1 2 5 6 0 7 1 7 6 0 6 0 5 8 8 6 1 1 6 4 6 7 1 0 9 4 0 5 0 7 7 5 4 1 0 0 2 2 5 6 9 8 3 1 5 5 2 0 0 0 5 5 9 3 5 7 2 9 7 2 5 7 1 6 3 6 2 6 9 5 6 1 8 8 2 6 7 0 4 2 8 2 5 2 4 8 3 6 0 0 8 2 3 2 5 7 5 3 0 4 2 0 7 5 2 9 6 3 4 5 0/;
for($i=0;$i<996;$i++)
{
$sum=$num[$i]*$num[$i+1]*$num[$i+2]*$num[$i+3]*$num[$i+4];
if($sum>$big)
{
$big=$sum;
}
else
{
next;
}
}
print $big."\n";
結果如下:
C:\WINDOWS\system32\cmd.exe /c perl "F:\perl\c.pl"
40824
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第九題:找出唯一滿足a+b+c=1000的畢達格斯三元組
A Pythagorean triplet is a set of three natural numbers, a b c, for which,
a2 + b2 = c2For example, 32 + 42 = 9 + 16 = 25 = 52.
There exists exactly one Pythagorean triplet for which a + b + c =
1000.
Find the product abc.
一個畢達哥拉斯三元組是一個包含三個自然數的集合,a<b<c,滿足條件:
a2 + b2 = c2例如:32 + 42 = 9 + 16 = 25 = 52.
已知存在並且只存在一個畢達哥拉斯三元組滿足條件a + b + c = 1000。
找出該三元組中abc的乘積。
程式如下:
use strict;
use warnings;
my $first;
my $second;
my $sum;
for($first=1;$first<333;$first++)
{
for($second=$first+1;$second<500;$second++)
{
if($first**2+$second**2==(1000-$first-$second)**2)
{
print "$first X $second X(1000-$first-$second)=";
$sum=$first*$second*(1000-$first-$second);
print $sum."\n";
last;
}
else
{
next;
}
}
}
結果如下:
C:\WINDOWS\system32\cmd.exe /c perl "F:\perl\c.pl"
200 X 375 X(1000-200-375)=31875000
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這個題目和前面求素數的問題類似,具體的方案就不說了,直接帖程式碼:
use strict;
use warnings;
my $i;
my $sum=2;
my $cout;
my $flag=0;
for($i=3;$i<2000000;$i+=2)
{
$flag=0;
for($cout=2;$cout<$i**0.5+1;$cout++)
{
if(0==$i%$cout)
{
$flag=1;
last;
}
}
$sum=$sum+$i if($flag==0);
}
print $sum."\n";
結果如下:
C:\WINDOWS\system32\cmd.exe /c perl "F:\perl\c.pl"
142913828922
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