Practice Quiz 1: The quiz will be open book (any book) and open notes (any notes or written material).
[1] Use Euclid's algorithm to write the greatest common divisor of 13 and 47 as an integer linear combination of 13 and 47.
[2] Suppose a developer builds houses painted in one of 7 colors, uses any of 5 floor plans, with either electric or gas heat.
(a) How many different houses does the developer build? Explain your answer.
(b) In a subdevelopment of 50 houses, must there be at least two identical houses? Explain why or why not.
(c) Suppose the development has 200 houses. What is the largest number of houses that you can be sure have to be identical knowing nothing more than your answer to (a) and that there are only 200 houses? Explain your answer.
[3] Write 38 as a sum of Fibonacci numbers, no two of which are consecutive Fibonacci numbers.
[4] John and Jane play a game of Fibonacci nim, starting with a total of 26. John goes first.
(a) How many should John take away in order to be sure to win? Explain your answer.
(b) If John takes away only 2, can Jane win? Explain.
[5] An arithmetic sequence of numbers is a sequence where the difference being consecutive numbers is always the same. For example, for 1, 2, 3, ..., 50, the difference is always 1 since each number is always 1 bigger than the previous number. For 4, 7, 10, ..., 70, 73, the difference is always 3, since each number is always 3 bigger than the previous number. Gauss's trick works to add up any arithmetic sequence of numbers. Recall how Gauss's trick works: write the sum and below it write it in reverse order.
1 + 2 + ... + 100
100 + 99 + ... + 1
Add up the numbers in columns (each column here gives 101), multiply by the number of columns (100*101=10100) and divide by 2, which gives in this case 5050.
(a) Use Gauss' trick to add up the whole numbers from 1 to 50.
(b) Use Gauss's trick to add up the whole numbers from 100 to 200.
(c) Use Gauss's trick to add up the whole numbers from 4 to 73, counting by 3's; i.e., add up
4 + 7 + 10 + ... + 73.