문제
길고 얇은 막대를 만드는 과정은 다음과 같습니다.
- 기본으로 사용할 막대를 고릅니다. 사용 가능한 막대의 길이는 $A_1, \cdots, A_N$입니다.
- 원하는 길이가 될 때까지 0ドル$번 이상 막대를 늘립니다. 기계에 2ドル$ 이상의 양의 정수 $k$를 설정해서 막대를 넣으면, 길이가 $x$였던 막대가 길이 $kx$의 막대가 됩니다.
주어진 $Q$개의 길이 각각에 대해 길이가 $L_i$인 막대를 만드는 방법의 수를 출력하세요. 두 방법이 서로 다르다는 것은 처음 선택한 막대가 다르거나, 막대를 늘리는 과정에서 입력한 정수 $k$의 수열이 서로 다르다는 것을 의미합니다.
출력
첫 줄에 $Q$개의 수를 공백으로 구분해 출력합니다. $i$번째 수는 길이가 $L_i$인 막대를 만드는 방법의 수입니다. 가능한 모든 입력에 대해 답이 10ドル^9$을 넘지 않음을 증명할 수 있습니다.
길이 6ドル$인 막대를 만드는 서로 다른 방법은 다음과 같습니다.
- 길이가 1ドル$인 막대를 고릅니다. 길이를 2ドル$배 늘려 길이를 2ドル$로 만듭니다. 길이를 3ドル$배 늘려 길이를 6ドル$으로 만듭니다.
- 길이가 1ドル$인 막대를 고릅니다. 길이를 3ドル$배 늘려 길이를 3ドル$로 만듭니다. 길이를 2ドル$배 늘려 길이를 6ドル$으로 만듭니다.
- 길이가 1ドル$인 막대를 고릅니다. 길이를 6ドル$배 늘려 길이를 6ドル$으로 만듭니다.
- 길이가 2ドル$인 막대를 고릅니다. 길이를 3ドル$배 늘려 길이를 6ドル$으로 만듭니다.
- 길이가 3ドル$인 막대를 고릅니다. 길이를 2ドル$배 늘려 길이를 6ドル$으로 만듭니다.
[{"problem_id":"28437","problem_lang":"0","title":"\ub9c9\ub300 \ub9cc\ub4e4\uae30","description":"<p>\uae38\uace0 \uc587\uc740 \ub9c9\ub300\ub97c \ub9cc\ub4dc\ub294 \uacfc\uc815\uc740 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\uae30\ubcf8\uc73c\ub85c \uc0ac\uc6a9\ud560 \ub9c9\ub300\ub97c \uace0\ub985\ub2c8\ub2e4. \uc0ac\uc6a9 \uac00\ub2a5\ud55c \ub9c9\ub300\uc758 \uae38\uc774\ub294 $A_1, \\cdots, A_N$\uc785\ub2c8\ub2e4.<\/li>\r\n\t<li>\uc6d0\ud558\ub294 \uae38\uc774\uac00 \ub420 \ub54c\uae4c\uc9c0 $0$\ubc88 \uc774\uc0c1 \ub9c9\ub300\ub97c \ub298\ub9bd\ub2c8\ub2e4. \uae30\uacc4\uc5d0 $2$ \uc774\uc0c1\uc758 \uc591\uc758 \uc815\uc218 $k$\ub97c \uc124\uc815\ud574\uc11c \ub9c9\ub300\ub97c \ub123\uc73c\uba74, \uae38\uc774\uac00 $x$\uc600\ub358 \ub9c9\ub300\uac00 \uae38\uc774 $kx$\uc758 \ub9c9\ub300\uac00 \ub429\ub2c8\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc8fc\uc5b4\uc9c4 $Q$\uac1c\uc758 \uae38\uc774 \uac01\uac01\uc5d0 \ub300\ud574 \uae38\uc774\uac00 $L_i$\uc778 \ub9c9\ub300\ub97c \ub9cc\ub4dc\ub294 \ubc29\ubc95\uc758 \uc218\ub97c \ucd9c\ub825\ud558\uc138\uc694. \ub450 \ubc29\ubc95\uc774 \uc11c\ub85c \ub2e4\ub974\ub2e4\ub294 \uac83\uc740 \ucc98\uc74c \uc120\ud0dd\ud55c \ub9c9\ub300\uac00 \ub2e4\ub974\uac70\ub098, \ub9c9\ub300\ub97c \ub298\ub9ac\ub294 \uacfc\uc815\uc5d0\uc11c \uc785\ub825\ud55c \uc815\uc218 $k$\uc758 \uc218\uc5f4\uc774 \uc11c\ub85c \ub2e4\ub974\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud569\ub2c8\ub2e4.<\/p>\r\n","input":"<p>\uccab \uc904\uc5d0 \ub9c9\ub300\uc758 \uac1c\uc218 $N$\uc774 \uc8fc\uc5b4\uc9d1\ub2c8\ub2e4. $(1 \\le N \\le 100\\,000)$<\/p>\r\n\r\n<p>\ub458\uc9f8 \uc904\uc5d0 $N$\uac1c\uc758 \uc11c\ub85c \ub2e4\ub978 \uc815\uc218 $A_1, A_2, \\cdots, A_N$\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9d1\ub2c8\ub2e4. $(1 \\le A_i \\le 100\\,000)$<\/p>\r\n\r\n<p>\uc14b\uc9f8 \uc904\uc5d0 \ub9cc\ub4e4\uace0\uc790 \ud558\ub294 \ub9c9\ub300 \uae38\uc774\uc758 \uac1c\uc218 $Q$\uac00 \uc8fc\uc5b4\uc9d1\ub2c8\ub2e4. $(1 \\le Q \\le 100\\,000)$<\/p>\r\n\r\n<p>\ub137\uc9f8 \uc904\uc5d0 $Q$\uac1c\uc758 \uc815\uc218 $L_1, \\cdots, L_Q$\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9d1\ub2c8\ub2e4. $(1 \\le L_i \\le 100\\,000)$<\/p>\r\n","output":"<p>\uccab \uc904\uc5d0 $Q$\uac1c\uc758 \uc218\ub97c \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ud574 \ucd9c\ub825\ud569\ub2c8\ub2e4. $i$\ubc88\uc9f8 \uc218\ub294 \uae38\uc774\uac00 $L_i$\uc778 \ub9c9\ub300\ub97c \ub9cc\ub4dc\ub294 \ubc29\ubc95\uc758 \uc218\uc785\ub2c8\ub2e4. \uac00\ub2a5\ud55c \ubaa8\ub4e0 \uc785\ub825\uc5d0 \ub300\ud574 \ub2f5\uc774 $10^9$\uc744 \ub118\uc9c0 \uc54a\uc74c\uc744 \uc99d\uba85\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean","sample_explain_1":"<p>\uae38\uc774 $6$\uc778 \ub9c9\ub300\ub97c \ub9cc\ub4dc\ub294 \uc11c\ub85c \ub2e4\ub978 \ubc29\ubc95\uc740 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\uae38\uc774\uac00 $1$\uc778 \ub9c9\ub300\ub97c \uace0\ub985\ub2c8\ub2e4. \uae38\uc774\ub97c $2$\ubc30 \ub298\ub824 \uae38\uc774\ub97c $2$\ub85c \ub9cc\ub4ed\ub2c8\ub2e4. \uae38\uc774\ub97c $3$\ubc30 \ub298\ub824 \uae38\uc774\ub97c $6$\uc73c\ub85c \ub9cc\ub4ed\ub2c8\ub2e4.<\/li>\r\n\t<li>\uae38\uc774\uac00 $1$\uc778 \ub9c9\ub300\ub97c \uace0\ub985\ub2c8\ub2e4. \uae38\uc774\ub97c $3$\ubc30 \ub298\ub824 \uae38\uc774\ub97c $3$\ub85c \ub9cc\ub4ed\ub2c8\ub2e4. \uae38\uc774\ub97c $2$\ubc30 \ub298\ub824 \uae38\uc774\ub97c $6$\uc73c\ub85c \ub9cc\ub4ed\ub2c8\ub2e4.<\/li>\r\n\t<li>\uae38\uc774\uac00 $1$\uc778 \ub9c9\ub300\ub97c \uace0\ub985\ub2c8\ub2e4. \uae38\uc774\ub97c $6$\ubc30 \ub298\ub824 \uae38\uc774\ub97c $6$\uc73c\ub85c \ub9cc\ub4ed\ub2c8\ub2e4.<\/li>\r\n\t<li>\uae38\uc774\uac00 $2$\uc778 \ub9c9\ub300\ub97c \uace0\ub985\ub2c8\ub2e4. \uae38\uc774\ub97c $3$\ubc30 \ub298\ub824 \uae38\uc774\ub97c $6$\uc73c\ub85c \ub9cc\ub4ed\ub2c8\ub2e4.<\/li>\r\n\t<li>\uae38\uc774\uac00 $3$\uc778 \ub9c9\ub300\ub97c \uace0\ub985\ub2c8\ub2e4. \uae38\uc774\ub97c $2$\ubc30 \ub298\ub824 \uae38\uc774\ub97c $6$\uc73c\ub85c \ub9cc\ub4ed\ub2c8\ub2e4.<\/li>\r\n<\/ul>\r\n"},{"problem_id":"28437","problem_lang":"1","title":"Making Sticks","description":"<p>The process of making long, thin sticks is as follows:<\/p>\r\n\r\n<ul>\r\n\t<li>You select a stick to use as a base backbone. The available stick lengths are $A_1, \\cdots, A_N$.<\/li>\r\n\t<li>You extend the stick $0$ or more times, until it reaches the desired length. If you enter a positive integer $k$ greater than or equal to $2$ to the machine and insert the stick in a machine, a stick with a length of $x$ becomes a stick of length $kx$.<\/li>\r\n<\/ul>\r\n\r\n<p>For each of the given $Q$ lengths, print the number of ways to make a stick of length $L_i$. Two methods are considered different if the initial stick selected is different or if the sequence of integers $k$ input in the process of extending the stick differs.<\/p>\r\n","input":"<p>The first line contains the number of base sticks $N$. $(1 \\le N \\le 100\\,000)$<\/p>\r\n\r\n<p>The second line contains $N$ different integers $A_1, A_2, \\cdots, A_N$, separated by spaces. $(1 \\le A_i \\le 100\\,000)$<\/p>\r\n\r\n<p>The third line contains the number of desired stick lengths $Q$. $(1 \\le Q \\le 100\\,000)$<\/p>\r\n\r\n<p>The fourth line contains $Q$ integers $L_1, \\cdots, L_Q$, separated by spaces. $(1 \\le L_i \\le 100\\,000)$<\/p>\r\n","output":"<p>The first line should output $Q$ numbers, separated by spaces. The $i$&#39;th number is the number of ways to make a stick of length $L_i$. It can be proven that the answer does not exceed $10^9$ for all possible inputs.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English","sample_explain_1":"<p>The distinct ways to make a stick of length $6$ are as follows:<\/p>\r\n\r\n<ul>\r\n\t<li>Select a stick of length $1$. Increase its length by $2$ times to make it $2$. Increase its length by $3$ times to make it $6$.<\/li>\r\n\t<li>Select a stick of length $1$. Increase its length by $3$ times to make it $3$. Increase its length by $2$ times to make it $6$.<\/li>\r\n\t<li>Select a stick of length $1$. Increase its length by $6$ times to make it $6$.<\/li>\r\n\t<li>Select a stick of length $2$. Increase its length by $3$ times to make it $6$.<\/li>\r\n\t<li>Select a stick of length $3$. Increase its length by $2$ times to make it $6$.<\/li>\r\n<\/ul>\r\n"}]