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$X$ and $Y$ are two-digit numbers. If $Y=2X+2$ and $Y=2X$ in decimal and octal system respectively, and unit digits of $X$ and $Y$ are $5$ and $2$ respectively, then how to find $X+Y$ in decimal number system?

My attempt:

I tried representing the two numbers in decimal as $(10a+5,10b+2)$ and in octal as $(8a+5,8b+2)$ and then tried to manipulate with according to the conditions $Y=2X+2$ and $Y=2X$, but they only give me one equation $b-2a=1$, how to get $10a+10b+7$(the sum of $X+Y$ in decimals) from these?

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    @GarouDan: $b$ it the leading digit of $Y$, so the relations are $10b+2=2(10a+5)+2$ and $8b+2=2(8a+5)$, which are redundant as MaX says.2011-11-11

2 Answers 2

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Do you mean "unit digits of $X$ and $Y$ are $5$ and $2$ respectively"? I agree with you that $a$ can be any of $1,2, \text{or } 3$ (no higher or $Y$ will carry in octal) and there is no single answer.

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Edit: (shortened) We are told that the (decimal) units digit of $X$ is $5$. So the only candidates for $X$ are $15$, $25$, and $35$. (Anything bigger, when expressed in octal, then doubled, is not a two-digit octal number.) The specification that $Y$ has units digit $2$ is superfluous.

Check which ones of $15$, $25$, and $35$ work. They all do.

For a problem in which the numbers are so nearly pinned down, trying to use "algebra" can be a waste of time. Before introducing symbols, it is useful to play with the numbers to get a concrete grip on the problem.

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    Tha$n$ks, and I think we have [recent question](http://math.stackexchange.com/questions/80918/) on the similar topic.2011-11-12