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positive numbers

(Not necessarily whole numbers). Explain why we can write a=m_1b+r1 where m_1 is a non-negative integer and 0 less than or equal to r_1 less than b. Why are m_1 and r_1 unique? Continuing in this fashion, we can next write b=m_2r_1+r_2 with 0 less than or equal to r_2 less than r_1 and m_2 a non-negative integer. This procedure can now be repeated (next step being r_1=m_3r_2+r_3) as often as necessary. If one of the remainders is 0 then the process terminates. If not, then it goes on forever. We would like to study how the sequence of numbers m_1, m_2,.. is related to the choice of a and b.

a) execute this procedure for a=21, b=15, a=117, b=49, and list the m_i's
b) explain why the procedure must terminate if a, b are whole numbers
c) show that the procedure terminates if and only if a/b is a rational number
d) find the sequences of m_i's for a=5, b=2, a=10, b=4, a=15, b=6. Does the sequence m_1, m_2, ...uniquely determine the value a/b?

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