When are 1n! repeating decimal with single digit repetend

Step 1A positive rational number has terminating decimal expansion iff it can be expressed as x=A10k for some A,k∈NA rational number has a length-1 periodic decimal expansion iff it's not in the form above, but 10x−x=9x=A10k.Then, if x=1m we require that m=2a5b3c with a≥0, b≥0 and c≥1That 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∈{3,4,5,6}

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High School
Algebra
Pre-Algebra
Decimals

fun9vk

Answered question

2022-03-15

When are

\frac{1}{n!}

repeating decimal with single digit repetend

Answer & Explanation

pintorreeqwf

Beginner2022-03-16Added 3 answers

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}

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