Prove that the decimal expression of any rational number is periodic...?

Question

Prove that the decimal expression of any rational number is periodic. For this, let a/b be your rational number, with (a, b) = 1 and b belongs to N.
(a) Show that there are two different non-negative integers k > l >= 0 such that (10^k − 10^l) is a multiple of b. Hint: if (b, 10) = 1 then apply Euler’s theorem. If not, write b = b1 x b2 with (10, b1) = 1 and b2 = 2r1 x 5r2 . Apply Fermat’s little theorem to b1 and figure out what to do with b2.
(b) Say why the previous part implies periodicity of the decimal expression. What can you say about the period?
(c) The decimal expression is said to be pure periodic if the periodic part starts “right after the dot”. For example, 43/6 = 7.16, 1/56 = 017857142 and 2/15 = 0.06 are not pure, while 10/11 = 0.90 and 1/37 = 0.027 are pure. Show that the decimal expression of a/b is pure if and only if (10, b) = 1

Answer 1

So where, specifically, is your problem? There seem to be several completely independent parts so I find it hard to believe you tried and found you can't do any of it.

Answer 2

quoting "Cauchy" 's remark on the the sourced thread (there are different proofs besides): enable z be a rational huge type, and enable p and q be integers such that p/q = z and q > 0 . with the aid of fact multiplying a repeating decimal with the aid of a persevering with won't exchange that that is repeating, that is sufficient to educate that one million/q has the two a terminating or repeating decimal growth. enable q* be q with all aspects of two and 5 got rid of, it is, q* 2^m 5^n = q for some naturals m and n, and a pair of and 5 do not divide q*. Then, 10 and q* are incredibly best. If q* = one million, then one million/q* has a terminating decimal growth. in any different case, enable s be the Euler phi functionality utilized to q*, it is, s is the style of integers between 0 and q* inclusive that are incredibly best to q*. Then, q* divides 10^s - one million (that could be a straight forward actuality of modular arithmetic), so one million/q* = r/(10^s - one million) for some integer r under 10^s - one million. Then, r/(10^s - one million) = r 10^{-s} + r 10^{-2s} + r 10^{-3s} + ..., so one million/q* has a repeating decimal growth. Now we upload lower back interior the aspects of two and 5. as quickly as we divide the decimal growth with the aid of two, we are in a position to as a replace divide it with the aid of 10 (which needless to say preserves the two periodicity and finite length) and then multiply with the aid of five (which lower back preserves periodicity and finite length). further, dividing with the aid of five is like dividing with the aid of 10 and multiplying with the aid of two, so it additionally preserves finite length and periodicity. We divide with the aid of two m circumstances and divide with the aid of five n circumstances to end that one million/q has the two a finite decimal growth or a repeating decimal growth. hence, p/q has a kind of two varieties. with the aid of fact z replaced into arbitrary, all rationals have the two a finite decimal growth or a repeating decimal growth.