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

Let T be the set of all natural numbers that can be written as some nonnegative number of 3’s plus some nonnegative number of 5’s. For example, 9 = 3 + 3 + 3 and 10 = 5+5 and 17 = 3+3+3+3+5 are all in T, but 4 is not. Determine T (with justification). 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Show that you can get every positive integer except 1, 2, 4 and 7. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let . Case 1: for some . In this case we write as a sum of 3's. Case 2: for some with . In this case we write as the sum of 3's and one 5. Case 3: for some with . In this case we write as the sum of 3's and two 5's. This shows that all numbers other than 1,2,4,7 belong to . It is easy to see that 1,2,4,7 are not in . 