Science:Math Exam Resources/Courses/MATH220/April 2005/Question 06/Solution 1

Let $x\in \mathbb {N}$.

Case 1: $x=3k$ for some $k\in \mathbb {N}$.

In this case we write $x$ as a sum of $k$ 3's.

Case 2: $x=3k+2$ for some $k\in \mathbb {N}$ with $k\geq 1$.

In this case we write $x=3k+2=3(k-1)+5$ as the sum of $k-1$ 3's and one 5.

Case 3: $x=3k+1$ for some $k\in \mathbb {N}$ with $k\geq 3$.

In this case we write $x=3(k-3)+10$ as the sum of $k-3$ 3's and two 5's.

This shows that all numbers other than 1,2,4,7 belong to $T$. It is easy to see that 1,2,4,7 are not in $T$.