When are \(\displaystyle{\frac{{{1}}}{{{n}!}}}\) repeating decimal with single

Question

When are

\frac{1}{n!}

repeating decimal with single digit repetend

Answer 1

Step 1
A positive rational number has terminating decimal expansion iff it can be expressed as

x = \frac{A}{10^{k}}

for some

A, k \in N

A rational number has a length-1 periodic decimal expansion iff it's not in the form above, but

10 x - x = 9 x = \frac{A}{10^{k}}

.
Then, if

x = \frac{1}{m}

we require that

m = 2^{a} 5^{b} 3^{c}

with

a \geq 0, b \geq 0

and

c \geq 1

That is, it must be a multiple of 3, and it can be a multiple of 2 and 5, with no other ' factor.
Then, if

m = n!

, we must have

n \in {3, 4, 5, 6}

Answers (1)