perplexus.info :: Just Math : Floor Ratio Crossed Last Two Digits

Floor Ratio Crossed Last Two Digits (Posted on 2023-09-21)

Determine the last two digits of this expression:

⌊(10^2222)/(10^774+3)⌋

Notes:
1) ⌊n⌋ is the floor of n, that is, the greatest integer less than or equal to n.
2) Computer program/excel solver assisted solutions are welcome, but a semi-analytic (p&p and hand calculator) methodology is preferred.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Computer Solution

| Comment 1 of 3

The last 2 digits are: 00

It looks like the exact answer for the greatest integer is:

a string of 773 9s

then a 7

then a string of 674 zeros

then the remainder is 9*10^674

--------------

numer = '1' + '0'*2222

denom = int('1' + '0'*773 + '3')

def large_divide(divisor, dividend):

if type(dividend) == str:

dividend = [int(i) for i in dividend]

elif type(dividend) == int:

dividend = [int(i) for i in str(dividend)]

quotient = []

tempdividend = 0

i=0

for i,v in enumerate(dividend):

tempdividend = tempdividend*10 + dividend[i]

quotient.append(tempdividend // divisor)

tempdividend = tempdividend % divisor

wholepart = ''.join([str(j) for j in quotient])

remainder = tempdividend

ans = wholepart.lstrip('0')+'\nwith remainder '+ str(remainder)

return ans

print(large_divide(denom, numer))

-------------- Output

999999999999999999999999999999999999999999999999999999

999999999999999997000000000000000000000000000000000000

000000000000000000000000000000000000000000000000000000

00000000000000000000000000000000000000000000

with remainder 900000000000000000000000000000000000000000000000000000

000000000000000000000000000000000000000000000000000000

000000000000000000000000000

Posted by Larry on 2023-09-21 13:10:48

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