Given two integers a and m larger than 1, show that, if m is odd, then then is a divisor of . Use this result to obtain the factorization of 1001.
Since m is odd, we have and the result follows.
This shows that
Generalize the result of the above problem to obtain that if a and m are two integers larger than 1 and if is an odd divisor of m, then is a divisor of . Use this result to show that 101 is a factor of 1000001.
The result is immediate if we write and then we apply the result of the above problem. This shows that
Show that 7, 11 and 13 are factors of
It follows from the previous result that .
The result then follows from the fact that 7, 11, and 13 factors of 1001.
Show that is a composite number for each integer . More generally, show that if a is a positive integer such that is a perfect square, then is a composite number provided that
It is enough to observe that . For the general case, we only need to observe that .
We also make an observation that the condition is sufficient but not necessary.