MATH220 April 2005
• Q1 • Q2 • Q3 (a) • Q3 (b) • Q3 (c) • Q4 • Q5 • Q6 • Q7 • Q8 • Q9 • Q10 • Q11 •
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).
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Show that you can get every positive integer except 1, 2, 4 and 7.
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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 .
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