Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if b = 2n5m for some n,m ≥ 0. Hint: Observe that r has a finite decimal expansion when 10Nr is an integer for some N ≥ 0 (and hence b divides 10N).
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if
b = 2n5m for some n,m ≥ 0.
Hint: Observe that r has a finite decimal expansion when 10Nr is an integer for some N ≥ 0 (and hence b divides 10N).
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