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Problem 1063. -- Carmichael Numbers

Carmichael Numbers

Time Limit: 1 Sec  Memory Limit: 64 MB
Submit: 14  Solved: 5


Certain cryptographic algorithms make use of big prime numbers. However, checking whether a big number is prime is not so easy. Randomized primality tests exist that offer a high degree of confidence of accurate determination at low cost, such as the Fermat test. Let a be a random number between 2 and n - 1, where n is the number whose primality we are testing. Then, n is probably prime if the following equation holds: an mod n = a If a number passes the Fermat test several times, then it is prime with a high probability. Unfortunately, there is bad news. Certain composite numbers (non-primes) still pass the Fermat test with every number smaller than themselves. These numbers are called Carmichael numbers. Write a program to test whether a given integer is a Carmichael number.


The input will consist of a series of lines, each containing a small positive number n ( 2 < n < 65, 000). A number n = 0 will mark the end of the input, and must not be processed.


For each number in the input, print whether it is a Carmichael number or not as shown in the sample output

Sample Input


Sample Output

The number 1729 is a Carmichael number.
17 is normal.
The number 561 is a Carmichael number.
1109 is normal.
431 is normal.



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