In this problem we will explore the decimal expansion of nat

In this problem we will explore the decimal expansion of nationals of the form 1/n, n N. where f (n) is the length of the recurring cycle in the decimal expansion of 1/n. Notice that if the decimal expansion of 1/n terminates then f(n) = 1 because of the classify all values of n where 1/n terminates. Find ul1 values of n so that f(n) - 1. ] Find all values of n so that

Solution

f(n) terminates for n = 2, 4, 8, 16, ...

or n = 5, 25, 5^3.... or

n =10, 10^2,......

f(n) will be terminating

b) .n = 3,6,9,12......f(n) =1

but non terminating.

In otherwords in powers of 2, or 5 or 10 have f(n) =1.

Hence all powers of 2, multiples of 3, multiples of 5, and 10 are such that

f(n) = 1

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c) f(n) = 2

11 and multiples of 11 have

f(n) =2

f(37) =3

37 and multiples of 37 have f(n) =3

d) For any natural number m we have to find a n such that f(n) = m

Since we already found out there are images for 1,2,3

f(13) =6

f(17) = 16

Thus it is possible to find an image for a natural number.

Hence f is surjective.