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What is the value of $142,857 \times 7^2$?

Obviously you could solve this with a calculator and be done. But is there a more clever way to calculate this?

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Do you recognize that $1/7=0.\overline{142857}?$ If so, you will recognize that $142,857 \cdot 7 = 999,999,$ so $142,857 \cdot 7^2=(1,000,000-1)\cdot 7=6,999,993$

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  • $\begingroup$ I figured it out by doing $\dfrac{1}{7} = 0.142857...$ then $1/7(10^6-\dfrac{1}{7})$ to get the $999,999$. Is that what you did? $\endgroup$ – John Ryan Dec 16 '15 at 2:19
  • $\begingroup$ Because when you start the repeat of the decimal again, you have just $1$ left, so the product of any number (not having a factor $2$ or $5$) and the repeat of the decimal is a string of $9$'s of the length of the repeat. Similarly for $1/13=0.\overline {076923}$ and $13 \cdot 76923=999,999$ $\endgroup$ – Ross Millikan Dec 16 '15 at 2:20
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    $\begingroup$ I usually explain it in the following way to my students: Letting $a = 1/7$. Then according to the repeating decimals, we have $10^6 a - a = 142857$ and therefore $10^6 - 1 = 142857 \cdot 7$. $\endgroup$ – Empiricist Dec 16 '15 at 2:21
  • $\begingroup$ @RossMillikan Can you elaborate? What do you mean by just 1 left? $\endgroup$ – John Ryan Dec 16 '15 at 2:29
  • $\begingroup$ If you think about doing long division of $7$ into $1$ to find the decimal, the first division goes $1$ time with remainder $3$. The second division goes $4$ times with remainder $2$. The remainders will all be distinct. When you finish the repeat, the remainder is $1$. That is what "causes" the repeat-you are back where you started doing the division. $\endgroup$ – Ross Millikan Dec 16 '15 at 2:42
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Multipling by $7^2=49$ is the same as multiplying by $50$ then subtracting one copy.

So $$142,857*7^2=142,857*50-142857$$. Now, multipling by 50 is the same as multiplying by 100 then dividing by 2. so we have $$(142,857*100)/2-142857$$. Multiplying by 100 is just adding 2 0s: $$14285700/2-142857$$ Dividing by 2 is easy to do by hand: $$7142850-142857$$ And finally, subtracting two numbers is easy to do by hand: $$6999993$$ Hopefully that's actually right, because I'm not checking :)

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If you recognize that those are the decimal digits of $1/7 = .142857142857142857...$, then you realize that $1000000/7$ must be 142857+0.142857, so 142857 must be (100,000 - 1)/7, and you're on your way.

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