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 Sep 18 comment How to evaluate last nine digits of $(2^{120} - 1) / 3$ without any intermediate value exceeding $2^{32}$ Nevertheless, it was a valid answer that met the don't exceed $2^{32}$ limit. Thank you. Sep 18 comment How to evaluate last nine digits of $(2^{120} - 1) / 3$ without any intermediate value exceeding $2^{32}$ Another blindingly obvious that I missed. Thanks! Sep 18 accepted How to evaluate last nine digits of $(2^{120} - 1) / 3$ without any intermediate value exceeding $2^{32}$ Sep 18 comment How to evaluate last nine digits of $(2^{120} - 1) / 3$ without any intermediate value exceeding $2^{32}$ Aw, now I feel stupid. "How absurdly simple!" as Watson would say. Sep 18 comment How to evaluate last nine digits of $(2^{120} - 1) / 3$ without any intermediate value exceeding $2^{32}$ How did you evaluate $\frac{280344576}{3} \equiv 93448192 \pmod{10^9}$ Sep 18 awarded Editor Sep 18 revised How to evaluate last nine digits of $(2^{120} - 1) / 3$ without any intermediate value exceeding $2^{32}$ added 24 characters in body; edited title Sep 18 comment How to evaluate last nine digits of $(2^{120} - 1) / 3$ without any intermediate value exceeding $2^{32}$ @HenningMakholm -- I indicated in subject but not in the message; I am only interested in the last nine digits of that expression. I will indicate so in the message as well. Sep 18 asked How to evaluate last nine digits of $(2^{120} - 1) / 3$ without any intermediate value exceeding $2^{32}$ Feb 22 awarded Teacher Nov 26 comment Is the number 100k+11 ever a square? Correction -- " ... and 89 are squares" should read " ... and 89 can be squares". Oct 30 comment How can I express the sum of $\sin a+\sin2a+\sin3a+\cdots+\sin(n-1)a$? @Alex -- The answer posted here is very similar to what Boris Novikov has posted. Look up Telescopic sum Oct 19 comment Calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using pen and paper You have an Arithmetico Geometric Series. Oct 19 awarded Commentator Oct 19 comment In high school statistics, why does it seem like equations come out of the sky " They will slaughter us in a nuclear war next week if we don't make our students [memorize a formula] " -- Awesome! :) Oct 10 comment Solve $4^{9x-4} = 3^{9x-4}$ Try this: Solve for $x$ and $y$ when $p ^ {2x + y} = q ^ {x + y + 1}$ where $p$ and $q$ are two integers that are relatively prime to each other. Oct 8 comment Fastest way to check if $x^y > y^x$? Which means one has more permutations to test (I agree that it would still be far more efficient than calculating $x ^ y$ or $y ^ x$. Oct 8 comment Fastest way to check if $x^y > y^x$? @Henry -- Did you mean "... 1 and $e$"? When $log$'s operand is less than one, it becomes negative and one has to be careful with inequalities with one side being negative and other being positive. Oct 5 comment Why does my calculator show $2^{-329} = 0?$ Michael Hardy -- My +1 goes to @AlvinWong whose comment explains why the typical calculator shows zero. Jul 30 answered What is the meaning of infinitesimal?