---
{"category_name":"easy","problem_code":"SUBMIMX","problem_name":"Substring Minimum Function","problemComponents":{"constraints":"- $1 \\leq T \\leq 2 \\cdot 10^5$\n- $1 \\leq N \\leq 10^6$\n- $0 \\leq M \\leq N$\n","constraintsState":true,"subtasks":"","subtasksState":false,"inputFormat":"- First line of input contains a single integer $T$, denoting the number of test cases. Then the description of the $T$ test case follows.\n- Each test case contains one line of input.\n- The first line contains two integers $N$ and $M$.","inputFormatState":true,"outputFormat":"For each test case, output a single line answer containing an integer.","outputFormatState":true,"sampleTestCases":{"0":{"id":1,"input":"2\n3 1\n7 2","output":"2\n7","explanation":"In the first test case, there exists only 3 binary strings of length $N$, having exactly $M$ occurrences of the character $1$.\n\nFor $S = 100$ value of $F(S) = 3$\n\nFor $S = 001$ value of $F(S) = 3$\n\nFor $S = 010$ value of $F(S) = 2$\n\nSo for string $S = 010$ we get a minimum value of $F(S) = 2$.","isDeleted":false}}},"video_editorial_url":"https://youtu.be/Y7V53WqAqEA","languages_supported":{"0":"CPP14","1":"C","2":"JAVA","3":"PYTH 3.6","4":"CPP17","5":"PYTH","6":"PYP3","7":"CS2","8":"ADA","9":"PYPY","10":"TEXT","11":"PAS fpc","12":"NODEJS","13":"RUBY","14":"PHP","15":"GO","16":"HASK","17":"TCL","18":"PERL","19":"SCALA","20":"LUA","21":"kotlin","22":"BASH","23":"JS","24":"LISP sbcl","25":"rust","26":"PAS gpc","27":"BF","28":"CLOJ","29":"R","30":"D","31":"CAML","32":"FORT","33":"ASM","34":"swift","35":"FS","36":"WSPC","37":"LISP clisp","38":"SQL","39":"SCM guile","40":"PERL6","41":"ERL","42":"CLPS","43":"ICK","44":"NICE","45":"PRLG","46":"ICON","47":"COB","48":"SCM chicken","49":"PIKE","50":"SCM qobi","51":"ST","52":"SQLQ","53":"NEM"},"max_timelimit":1,"source_sizelimit":50000,"problem_author":"kaushal_26","problem_tester":"","date_added":"12-11-2021","tags":{"0":"fzbz2021","1":"fzbz2021","2":"kaushal_26","3":"math","4":"math","5":"simple","6":"simple"},"problem_difficulty_level":"Unavailable","best_tag":"","editorial_url":"https://discuss.codechef.com/problems/SUBMIMX","time":{"view_start_date":1637951400,"submit_start_date":1637951400,"visible_start_date":1637951400,"end_date":1735669800},"is_direct_submittable":false,"problemDiscussURL":"https://discuss.codechef.com/search?q=SUBMIMX","is_proctored":false,"visitedContests":{},"layout":"problem"}
---
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Tracy is teaching Charlie maths via a game called $N$-Cube, which involves three sections involving $N$.
Tracy gives Charlie a number $N$, and Charlie makes a list of $N$-th powers of integers in increasing order $1^N, 2^N, 3^N, \dot, \text{so on}$. This teaches him exponentiation.
Then Charlie performs the following subtraction game $N$ times: Take all pairs of consecutive numbers in the list and take their difference. These differences then form the new list for the next iteration of the game. Eg, if $N$ was 6, the list proceeds as $[1, 64, 729, 4096 ... ]$ to $[63, 685, 3367 ...]$, and so on $5$ more times.
After the subtraction game, Charlie has to correctly tell Tracy the $N$-th element of the list. This number is the *value of the game*.
After practice Charlie became an expert in the game. To challenge him more, Tracy will give two numbers $M$ (where $M$ is a prime) and $R$ instead of just a single number $N$, and the game must start from $M_R - 1$ instead of $N$. Since the *value of the game* can now become large, Charlie just have to tell the largest integer $K$ such that $M_K$ divides this number. Since even $K$ can be large, output $K$ modulo 1000000007 ($10^9 + 7$).
<aside style='background: #f8f8f8;padding: 10px 15px;'><div>All submissions for this problem are available.</div></aside>