Periods of repeating decimals!

Number Theory Level 3

${A = 0.\overline{19} + 0.\overline{199}, \quad B = 0.\overline{19} \times 0.\overline{199}}$

Recall that $0.\overline{19},$ for example, stands for the repeating decimal $0.19191919...$ and that the period of a repeating decimal is the number of digits in the repeating part. In this case, the period of $0.\overline{19}$ is 2.

Find the sum of the periods of $A$ and $B$ .

The answer is 60.

3 solutions

Mathh Mathh
Aug 7, 2015

The period of $A$ is $6$ , because $191919+199199=39118$ and $\text{lcm}(2,3)=6$ .

The period of $B$ is $54$ because:

$B=\frac{19}{99}\cdot\frac{199}{999}=\frac{3781}{98901}$

and the least $k\in\Bbb Z_{\ge 1}$ such that $98901n=10^k-1$ for some $n\in\Bbb Z_{\ge 1}$ is $54$ .

It's because $98901=3^5\cdot 11\cdot 37$ and $\text{lcm}\left(\text{ord}_{3^5}(10),\text{ord}_{11}(10),\text{ord}_{37}(10)\right)=\text{lcm}\left(27,2,3\right)=54$ .

To see why $\text{ord}_{3^5}(10)=27$ , apply Binomial theorem:

$(1+9)^k\equiv 1+9k+\frac{81k(k+1)}{2}\equiv 1\pmod{\! 3^5}$

$\iff 3^5\mid 81k^2+99k\iff 3^3\mid k(9k+11)$ .

Since $3\nmid 9k+11$ , we have $\iff 3^3\mid k$ , with the lowest positive $k$ being $27$ .

na verdade a resposta é 59 pois o periodo de B é 53. abaixo está o resultado em decimal de B 1,0382301493 4160452371 5634826745 9378570489 6816007927 119 03

Joel Rodrigues - 5 years, 4 months ago

can you please explain me what does "ord " means?

Charuka Bandara - 4 years, 10 months ago

"ord" denotes multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order). If $n\in\mathbb Z^+, n\ge 2$ and $a\in\mathbb Z, \gcd(n,a)=1$ , then $\text{ord}_n(a)$ denotes the least positive integer $k$ such that $a^k\equiv 1\pmod{n}$ .

mathh mathh - 4 years, 10 months ago

Thanks a lot

Charuka Bandara - 4 years, 10 months ago

ordinal :)

Jules van Praag - 3 years, 2 months ago

Where would I find a free online course that would include practicing calculating the order of two such numbers as these?

John Williamson - 4 years, 10 months ago

The binomial theorem doesn't account for distinction between primes and squares that can be represented by only the difference of two other squares

Ryan Matthew - 3 years, 4 months ago

i.e. 3 and 9, 3 and (5^2-4^2)

Ryan Matthew - 3 years, 4 months ago

I think period of B is 36 as (10^36-1)/(999*99) is an integer. So answer should be 36+6 = 42

Zakir Dakua - 5 years, 5 months ago

Sorry, may be not.

Zakir Dakua - 5 years, 5 months ago

Bill Bell
Aug 5, 2015

Examine the output from this code.

## http://code.activestate.com/recipes/578939-division-digit-by-digit-calculation/

def div(a,b,t):

    a1=str(a)
    if  "." in  a1:
        d1, d2 =a1.split(".")
        d2=d2.ljust(t,"0")    
    else:
        d1=a1
        d2="0"*t

    c=[];r1=0
    for i in d1:
            d11=int(str(r1)+i)
            c1=d11/b
            r1=d11%b
            c.append(c1);

    c.append(".")    
    for i in d2:
            d11=int(str(r1)+i)
            c1=d11/b
            r1=d11%b
            c.append(c1);


    c = "".join(map(str, c))
    while c[0] =="0": c =c[1:]
    if c[0]==".": c ="0"+c 

    return c 

from fractions import Fraction

a=Fraction(19,99)
b=Fraction(199,999)
A=a+b
B=a*b
print div(A.numerator,A.denominator,100)
print div(B.numerator,B.denominator,100)

Fractions are straight baller

Ryan Matthew - 3 years, 4 months ago

Soumik Pal
Nov 2, 2019

I am just explaining the crucial step of the sum in the easiest way. Work out the solution on your own.

0.19 stands for the repeating decimal (= 19/99):

0.1919191919191919191919191919191919191919191919191919…

and 0.199 for (= 199/999)

0.1991991991991991991991991991991991991991991991991991…

0.19 + 0.199 = 19/99 + 199/999 = 391118/999999

that is,

0.391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118391118…

Repeating decimal: 0.391118 (period 6)

0.19 * 0.199 = 19/99 * 199/999 = 3781/98901

3781/98901 = 3780/98901 + 1/98901

3780/98901 = 140/3663

0.038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220038220…

Repeating decimal: 0.0382200 (period 6)

And 1 / 98901 = is interesting:

0.00001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455556566667677778788889900001011112122223233334344445455…

0.0000101111212222323333434444545555656666767777878888990000 (period 54)

Can you figure out the solution now?

Periods of repeating decimals!

The answer is 60.

3 solutions

1 pending report

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