Problem 1) Show that the number is a multiple of 7.
Answer 1) If n is an odd positive integer, we know that . Therefore, , , and the result follows.
Problem 2) Compute the value of the expression .
Answer 2) (American Invitational Mathematics Examination) Let N be the number to compute. Since and since , then the number N can be written as
But, since , the number N can be written as
Problem 3) Given integers and a prime number p, show that p divides the integer
if and only if p divides , for an integer r, . Use this to find all integers n such that 7 divides .
Answer 3) Let , . Using the binomial theorem, we obtain
for a certain integer M. Hence, if and only if .
Setting , , we find that the required integers are those of the form as well as those of the form .
Problem 4) Show that there exist infinitely many pairs of integers satisfying and .
Let a and b be two arbitrary integers, and put , and . In order to have , we must have . Moreover, to have , we must have . Therefore, it remains to show that it is possible to find infinitely many relatively prime pairs of integers a and b such that . To do so, it is enough to choose, for example, and , where .
Problem 5) Prove that one cannot find integers m and n such that and .
This follows from the fact that and , while 3 does not divide 101.