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문제

x에 x를 30번 곱하면 x31이 된다.

x2 = x × x, x3 = x2 × x, x4 = x3 × x, ..., x31 = x30 × x

만약, 결과를 제곱할 수 있다면, 8번만에 x31을 구할 수 있다.

x2 = x × x, x3 = x2 × x, x6 = x3 × x3, x7 = x6 × x, x14 = x7 × x7, x15 = x14 × x, x30 = x15 × x15, x31 = x30 × x

이전에 나온 계산 결과를 곱하는 방법도 같이 사용한다면 x31은 7번만에 구할 수 있다.

x2 = x × x, x4 = x2 × x2, x8 = x4 × x4, x10 = x8 × x2, x20 = x10 × x10, x30 = x20 × x10, x31 = x30 × x

위의 방법이 곱셈만으로 x31을 구하는 가장 효율적인 방법이다.

만약 나눗셈을 사용할 수 있다면, 연산의 수를 더 줄일 수 있다. x31을 5번의 곱셈과 1번의 나눗셈으로 구할 수 있다.

x2 = x × x, x4 = x2 × x2, x8 = x4 × x4, x16 = x8 × x8, x32 = x16 × x16, x31 = x32 ÷ x

이 방법은 나눗셈이 곱셈만큼 빠를 때 x31을 계산하는 가장 효율적인 방법이다.

x로 시작해서 xn을 만드는데 드는 연산 회수의 최솟값을 구하는 프로그램을 작성하시오. 문제에서 설명한 곱셈과 나눗셈만 사용할 수 있다. 연산의 결과는 항상 x의 양의 제곱수 이어야 한다. 즉, x-3은 나오면 안 된다.

입력

첫째 줄에 테스트 케이스의 개수 T가 주어진다. 각 테스트 케이스는 한 줄로 구성되어 있고, 정수 n이 주어진다. n은 양의 정수이며, 1000보다 작거나 같다.

출력

각각의 테스트 케이스에 대해서, 한 줄에 하나씩 xn을 만드는데 필요한 곱셈과 나눗셈의 최소 회수를 출력한다.

예제 입력 1

8
1
31
70
91
473
512
811
953

예제 출력 1

0
6
8
9
11
9
13
12
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The following is a way to compute x<sup>31<\/sup> with eight multiplications:<\/p>\r\n\r\n<p>x<sup>2<\/sup>&nbsp;= x &times; x, x<sup>3<\/sup>&nbsp;= x<sup>2<\/sup>&nbsp;&times; x, x<sup>6<\/sup>&nbsp;= x<sup>3<\/sup>&nbsp;&times; x<sup>3<\/sup>, x<sup>7<\/sup>&nbsp;= x<sup>6<\/sup>&nbsp;&times; x, x<sup>14<\/sup>&nbsp;= x<sup>7<\/sup>&nbsp;&times; x<sup>7<\/sup>, x<sup>15<\/sup>&nbsp;= x<sup>14<\/sup>&nbsp;&times; x, x<sup>30<\/sup>&nbsp;= x<sup>15<\/sup>&nbsp;&times; x<sup>15<\/sup>, x<sup>31<\/sup>&nbsp;= x<sup>30<\/sup>&nbsp;&times; x<\/p>\r\n\r\n<p>This is not the shortest sequence of multiplications to compute x<sup>31<\/sup>. There are many ways with only seven multiplications. The following is one of them:<\/p>\r\n\r\n<p>x<sup>2<\/sup>&nbsp;= x &times; x, x<sup>4<\/sup>&nbsp;= x<sup>2<\/sup>&nbsp;&times; x<sup>2<\/sup>, x<sup>8<\/sup>&nbsp;= x<sup>4<\/sup>&nbsp;&times; x<sup>4<\/sup>, x<sup>10<\/sup>&nbsp;= x<sup>8<\/sup>&nbsp;&times; x<sup>2<\/sup>, x<sup>20<\/sup>&nbsp;= x<sup>10<\/sup>&nbsp;&times; x<sup>10<\/sup>, x<sup>30<\/sup>&nbsp;= x<sup>20<\/sup>&nbsp;&times; x<sup>10<\/sup>, x<sup>31<\/sup>&nbsp;= x<sup>30<\/sup>&nbsp;&times; x<\/p>\r\n\r\n<p>There however is no way to compute x<sup>31<\/sup> with fewer multiplications. Thus this is one of the most efficient ways to compute x<sup>31<\/sup> only by multiplications.<\/p>\r\n\r\n<p>If division is also available, we can find a shorter sequence of operations. It is possible to compute x<sup>31<\/sup> with six operations (five multiplications and one division):<\/p>\r\n\r\n<p>x<sup>2<\/sup>&nbsp;= x &times; x, x<sup>4<\/sup>&nbsp;= x<sup>2<\/sup>&nbsp;&times; x<sup>2<\/sup>, x<sup>8<\/sup>&nbsp;= x<sup>4<\/sup>&nbsp;&times; x<sup>4<\/sup>, x<sup>16<\/sup>&nbsp;= x<sup>8<\/sup>&nbsp;&times; x<sup>8<\/sup>, x<sup>32<\/sup>&nbsp;= x<sup>16<\/sup>&nbsp;&times; x<sup>16<\/sup>, x<sup>31<\/sup>&nbsp;= x<sup>32&nbsp;<\/sup>&divide; x<\/p>\r\n\r\n<p>This is one of the most efficient ways to compute x<sup>31<\/sup> if a division is as fast as a multiplication.<\/p>\r\n\r\n<p>Your mission is to write a program to find the least number of operations to compute x<sup>n<\/sup> by multiplication and division starting with x for the given positive integer n. Products and quotients appearing in the sequence of operations should be x to a positive integer&#39;s power. In other words, x<sup>-3<\/sup>, for example, should never appear.<\/p>\r\n","input":"<p>The input is a sequence of one or more lines each containing a single integer n. n is positive and less than or equal to 1000. The end of the input is indicated by a zero.<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","output":"<p>Your program should print the least total number of multiplications and divisions required to compute x<sup>n<\/sup> starting with x for the integer n. The numbers should be written each in a separate line without any superfluous characters such as leading or trailing spaces.<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"English"}]

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ICPC > Regionals > Asia Pacific > Japan > Asia Regional Contest 2006 in Yokohama F번