In this paper we are concerned with the following four types of integer problems. Integer Selection Problem (ISP): Let 
be positive integers and let 
and 
be given integers. Find an integer 
with 
and non-negative integers 
satisfying 
and 
for 
. 3-valued Problem: Given integers 
and 
find 
satisfying 
. Indeterminate Coefficient Problem (IDCP): Let m, n, p and q be positive integers, 
integers, and let 
and 
be given integers. Find non-negative integers 
and 
satisfying 
and 
. IDCP with boundedness conditions: Given integers 
, solve the IDCP under the boundedness conditions 
. The main objective in this paper is to show that the two problems ISP and IDCP are equivalent, and that any IDCP with boundedness conditions is reduced to a 3-valued problem, as stated below: Any solution of a given ISP (resp. IDCP and IDCP with boundedness conditions) is derived from solutions of the associated IDCP (resp. ISP and 3-valued problem) and vice versa. In order to state the second result on the 3-valued problem for given integers 
and 
, we introduce two notions: We say that a subset 
of 
is weakly (resp. strongly) removable, if for each 
there exists 
satisfying 
(resp. 
) and 
and we say that 
is maximal if 
for any 
is not weakly (resp. strongly) removable. Using the above terminologies, we may state the following result which provides useful necessary conditions for the existence of solutions of 3-valued problems: Let 
be a solution of the 3-valued problem formulated for 
and 
. (i) If there are no solutions 
such that 
for k with 
, then the subset 
is maximal as a weakly removable set and also maximal as a strongly removable set. (ii) If 
for any 
and the inequality 
holds for a subset 
of 
, than 
for some .gif)
.
