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
.
Solution 4:
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
.
Solution 5:
This follows from the fact that and
, while 3 does not divide 101.
More later,
Nalin Pithwa