What is the decimal representation of #17/5210# ?

2 Answers
Oct 13, 2015

Answer:

#17-:5210=.003262955854#

Explanation:

Using a calculator, #17-:5210=.003262955854#.

Oct 13, 2015

Answer:

#17/5210 = 0.0bar(0326295585412667946257197696737044145873320537428023)#

Explanation:

#521# is prime and is a factor of #10^52 - 1#.

#10^52 - 1 = 521 * 19193857965451055662188099808061420345489443378119#

So the decimal expansion of #1/521# repeats every #52# digits.

Hence #17 / 5210# will also repeat every #52# digits, after an initial extra #0# after the decimal point due to the factor of #10# in #5210#.

If you long divide #17 / 5210# you will notice that the remainder becomes #17# every #52# steps.

If you divide any positive integer by another, the result will be a decimal expansion that eventually repeats (including the case of repeating #0# - i.e. terminating), since the running remainder must eventually repeat a previous value.