That parenthetical remark is essential. Consider n=13. This has solutions for a,b in the integers. For example, a,b = -1,2 and 7,-1 are both solutions to 3a+8b=13. In fact, given any integer n, there are an infinite number of solutions a,b in Z such that 3a+8b=n. So the problem is to not only show that integer solutions a,b exist but also to show that at least one such solution has both a and b non-negative.Mark44 said:IOW, that n = 3a + 8b for some integer values of a and b, with a >= 0, b >= 0. (I don't think that either a or b can be negative, but either one could be zero.)
The problem statement is to prove that every integer n greater than or equal to 14 can be expressed as a sum of 3's and/or 8's.
2.This problem is important because it demonstrates the concept of integer compositions and can have real-world applications in areas such as computer science and number theory.
3.The number 14 is significant because it is the first integer that requires both 3's and 8's to form a sum. It serves as a starting point for the proof.
4.The approach to proving this statement is by using mathematical induction. This involves proving a base case (n=14) and then proving that if the statement holds for n=k, it also holds for n=k+1.
5.Integer compositions are ways of expressing an integer as a sum of smaller positive integers. For example, the integer 10 can be composed as 5+3+2, 4+4+2, or 3+3+3+1. This concept is important in this problem because we are trying to show that every integer n>=14 can be composed using only 3's and/or 8's.