Question about divisibility in different bases?

romsek

MHF Helper
Nov 2013
6,665
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California
9 and 10 have no common factors so any divisibility by 10 is going to come from the digits themselves.
This means we're going to need the digit 5 and an even digit to both appear. The digit 5 does not appear.
Thus this number base 9 is not divisible by 10.
 
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Plato

MHF Helper
Aug 2006
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This is a (necro)post that I'm bumping into a new thread.
-Dan
Dan, does "Is (447836)_9 divisible by 10 " mean that the base 10 equivalent of \(\displaystyle 447836_9\) is divisible by 10?
If so \(\displaystyle 447836_9=268224_{10}\) see here
Sorry I just fail to see the point of the question.
I lived through one technology revolution in which much of traditional topics were rendered obsolete.
Think: learning to graph functions, learning to use partial fractions, learning to sum infinite series, and on and on.
What is now important to know is what each of those implies.
 

topsquark

Forum Staff
Jan 2006
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Wellsville, NY
Dan, does "Is (447836)_9 divisible by 10 " mean that the base 10 equivalent of \(\displaystyle 447836_9\) is divisible by 10?
If so \(\displaystyle 447836_9=268224_{10}\) see here
Sorry I just fail to see the point of the question.
I lived through one technology revolution in which much of traditional topics were rendered obsolete.
Think: learning to graph functions, learning to use partial fractions, learning to sum infinite series, and on and on.
What is now important to know is what each of those implies.
That's a good question and the answer is that I don't know. This is the exact copy that Yagnesh made in the old post. I told him I reposted for him but I haven't heard back from him yet. Here's the original thread... He's got the last post.

-Dan
 

ChipB

MHF Helper
Jun 2014
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NJ
9 and 10 have no common factors so any divisibility by 10 is going to come from the digits themselves.
This means we're going to need the digit 5 and an even digit to both appear. The digit 5 does not appear.
Thus this number base 9 is not divisible by 10.
I'm not following this argument. I can think of numbers in base 9 that don't have 5 as a digit but whose base 10 equivalent are divisible by 10. For example (88)_9 is equal to (80)_10. And if you add 6 to Yagnesh's original number you get 447843, whose base 10 equivalent is divisible by ten. The only short cut I can think of to solving this problem (other than going through the tedious exercise of converting the number to base 10, as Plato did) is to examine the units digit that you get when converting each digit from base 9 to base 10, then add them and see if the result ends in 0. Here you could add (going from least significant to most) the units digits for 6x1 + 3x9 + 8x1 + 7x9 + 4x1 + 4x9, which results in 6 + 7 + 8 + 3 + 4 + 6 = 34, which is not divisible by 10.