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$112,263,147,00a$ is divisible by $18.$ Find the possible values of $a.$

To solve this I factored $18$ into $2$ and $9$, then I used the rule of $2$ and the rule of $9$.

By the rule of $2$, a may equal $0,2,4,6,8$

By the rule of $9$ we sum the digits and see if the result is divisible by $9.$ Summing the digits I have $27+a$, so a may equal $0$ or $9$

So, the only digit that follows both the rule of $2$ and $9$ is $0$.

However, when checking this with a calculator, we see that both $0$ and $9$ would work for this number to be divisible by $18.$ What have I done wrong?

user401699
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MathGuy
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  • If $a=9$ the number is clearly odd... – lulu Jan 06 '17 at 18:09
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    An even number cant divide an odd number. – 8hantanu Jan 06 '17 at 18:10
  • no since $$9$$ is an odd number your given number can not divided by two (without a remainder!) – Dr. Sonnhard Graubner Jan 06 '17 at 18:11
  • With $a=9$, the number is clearly not divisible by $18$ (as the number is odd). – lulu Jan 06 '17 at 18:11
  • Yes, I see. Maybe my calculator is not powerful enough for this. My TI-84 shows that this number equals 6236841501. Yes, dumb question but my calculator deceived me... Can't believe it isn't strong enough to handle this – MathGuy Jan 06 '17 at 18:13
  • If you have a PC or Mac with Python on it, try using Python as your calculator. You won't run out of the needed precision. – ncmathsadist Jan 06 '17 at 18:14
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    "When checking with a calculator... what have I done wrong" It seems what you've done wrong is trusted technology too much. Perhaps your calculator cannot handle such large numbers correctly. If you are using a simple four-function calculator with limited memory, it might not be able to tell the difference between $6236841500.\color{red}{5}$ and $6236841500.\color{red}{0}$. – JMoravitz Jan 06 '17 at 18:14
  • @JMoravitz no question I have trusted technology too much, but I must say that I'm very disappointed that a TI-84 graphing calculator cannot handle this... – MathGuy Jan 06 '17 at 18:15
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    You might also be using language that does integer division. For example in python 2 one can see that $1/2 = 0$. – Piotr Benedysiuk Jan 06 '17 at 18:15
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    Often-times in elementary-number-theory courses, the teachers will specifically choose numbers and examples that are too large to use calculators or software for in order to encourage strictly analytical solutions instead of brute force solutions. – JMoravitz Jan 06 '17 at 18:16
  • @TyeCampbell TI-84 is such a weird relic of "old times". Why schools still force the students to buy it, in this day and age is beyond me. – Piotr Benedysiuk Jan 06 '17 at 18:17
  • @PiotrBenedysiuk True. Kind of embarrassing that I asked this question as obviously 18 cannot divide an odd number. I've trusted technology way too much. I put this on one of my quizzes for my 8th graders and thought that my key was incorrect. – MathGuy Jan 06 '17 at 18:20
  • $112,263,147,009 \div 18$ gives you a whole number on a calculator???? What caluslator are you using. I get $112,263,147,009 \div 18=6236841500.5$. Obviously, as you reasoned, $112,263,147,009 $ is odd and can't be divisible by 18. – fleablood Jan 06 '17 at 18:22
  • fyi: What you call the "rule of 9" is widely known as casting out nines. – Bill Dubuque Jan 06 '17 at 18:23
  • " Maybe my calculator is not powerful enough for this. My TI-84 shows that this number equals 6236841501" which if you mulitple by 18 will clearly have last digit of 8. That is one crappy calculator! – fleablood Jan 06 '17 at 18:24
  • @fleablood yes, TI-84 Plus gives 6236841501. – MathGuy Jan 06 '17 at 18:25

1 Answers1

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(to remove question from unanswered queue)

You have done nothing wrong except trusted technology too much. The calculator you used has a limited memory when it comes to performing calculations and so incorrectly displayed $112263147009\div 18$ as an integer.

Of course, $112263147009$ being odd could not have an even number as a divisor. Your analytical solution was indeed correct and $0$ is the only choice for the value of $a$.

JMoravitz
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