Problem 16:Power digit sum

原题链接:http://projecteuler.net/problem=16

215 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.

What is the sum of the digits of the number 21000?


题目大意是:

215 = 32768 并且其各位之和为 is 3 + 2 + 7 + 6 + 8 = 26.

21000 的各位数之和是多少?


解法1:

由于21000太大了,不能直接相乘而得出乘积,按位相乘并进位则能较快得出结果。例如,计算1024*2的乘积时,是用2分别乘以4,2,0,1,有进位则进位。

php代码:

$base = 2;
$exp = 1000;

$arr = str_split($base);

$count = count($arr);
$new_count = $count;
$max_digit_count = $count*$exp-$exp+1;
$prev_result = $arr;
$new_result = array();

for($i=1;$i<$exp;$i++){
	$new_result = array_fill(0,$max_digit_count,0);
	for($j=0;$j<$count;$j++){		
		$over = 0;
		for($k=0;$k<$new_count;$k++){
			$temp_sum = $arr[$j]*$prev_result[$k]+$over;
			$pos = $j+$k;
			if(!isset($new_result[$pos])){
				$new_result[$pos] = 0;
			}
			if(!isset($new_result[$pos+1])){
				$new_result[$pos+1] = 0;
			}
			$new_result[$pos] += $temp_sum%10;
			$new_result[$pos+1] += intval($new_result[$pos]/10);
			$new_result[$pos] %= 10;
			$over = intval($temp_sum/10);
		}

		if(!isset($new_result[$pos+2])){
				$new_result[$pos+2] = 0;
		}
		$new_result[$pos+1] += $over;
		$new_result[$pos+2] += intval($new_result[$pos+1]/10);
		$new_result[$pos+1] %= 10;
		
	}
	
	$new_count = $j+$k;
	$prev_result = array_slice($new_result,0,$new_count);
}

echo array_sum($new_result);


注:题目的中文翻译源自http://pe.spiritzhang.com

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