TIP1 - Totient in permutation (easy)

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In number theory, Euler's totient (or PHI function), is an arithmetic function that counts the number of positive integers less than or equal to a positive integer N that are relatively prime to this number N.

That is, if N is a positive integer, then PHI(N) is the number of integers K for which GCD(N, K) = 1 and 1 ≤ K ≤ N. We denote GCD the Greatest Common Divisor. For example, we have PHI(9)=6.

Interestingly, PHI(87109)=79180, and it can be seen that 87109 is a permutation of 79180.


The input begins with the number T of test cases in a single line.
In each of the next T lines there are an integer M.


For each given M, you have to print on a single line the value of N, for which 1 < N < M, PHI(N) is a permutation of N and the ratio N/PHI(N) produces a minimum. If there's several answers output the greatest, or if need, "No solution." without quotes.
Leading zeros are not allowed for integers greater than 0.




Explanations : For the first case, in the range ]1..22[, the lonely number n for witch phi(n) is in permutations(n) is 21, (we have phi(21)=12). So the answer is obviously 21.
For the second case, in the range ]1..222[, there's two numbers n for witch phi(n) is in permutations(n), we have phi(21)=12 and phi(63)=36. But as 63/36 is equal to 21/12, we're taking the greater : 63.
For the third case, in the range ]1..2222[, there's four numbers n for witch phi(n) is in permutations(n), phi(21)=12, phi(63)=36, phi(291)=192 and phi(502)=250. Within those solutions 291/192 is the minimum, we output 291.


1 < T < 10^5
1 < M < 10^7

Code size limit is 10kB ; less than 500B of python3 code can get AC under 2s.
After that you may try TIP2.
@Speed addicts : my C code ran in 0.02s, and my fastest python3.2 code ran in 1.21s, (0.90s in py2.7)

Edit 2017-02-11, after compiler updates. My old C code ends in 0.00s, my old Python code ends in 0.05s !!!

hide comments
naim6246: 2020-08-19 01:50:54

Can anyone plz tell me the approach for this problem?

aman15: 2020-08-12 17:21:30

please anyone tell me what are the approach to solve this problem??

dip_1703042: 2019-12-20 18:41:51

anyone please help me to find the logic of the problem

selfcoder24: 2019-07-14 16:21:49

@Francky Can you please check my code is getting WA ??
Submission ID 24075839
Finally solved it mind that n<m

Last edit: 2019-07-14 17:09:27
priyanshu_pg: 2019-01-09 08:56:13

Thanks @Francky.
Mind n<m

priyanshu_pg: 2019-01-08 12:25:41

@Francky Can you please check my code. I am getting right answers for every test case on Spoj Toolkit.

=(Francky)=> S-Toolkit isn't official and maybe have wrong or incomplete answers... You have wrong answers for small inputs.

Last edit: 2019-01-08 15:48:33
Francky: 2017-02-12 15:36:17

Congratulations to Tjandra for being the lonely (for now) Python solver. Here, there's a little challenge.

DHEERAJ KUMAR: 2016-03-20 22:47:29

@Francky do i have WA for many inputs? submission id 16561400

.:frUstrAteD:.: 2015-10-03 21:50:00

@Francky Can you please check my code is getting WA
=(Francky)=> You have WA for only some small values. Good luck.

Last edit: 2015-10-12 21:02:36
Maroof: 2015-07-12 13:30:47

@FRANCKY do I have a lot of WAs in my last submission?

For small input, you have lot of WA, for bigger ones, it seems OK. I just gave a quick look. Have fun.

Last edit: 2015-07-12 13:52:02

Added by:Francky
Time limit:1.399s
Source limit:10000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ASM64
Resource:Extension of Project Euler n°###