In what bases, b, does (b + 10) divide into (8b + 10) without any remainder?

jamesbroadman Feb 27, 2020

#2**+1 **

Are those expressions in base b or in base 10?

If it is in base b then b=10 so b+10 = 2*b base b

and 8b+10 = 8b+b = 9b base b

But if the expressions are in base 10 then b+10 = 10+b base 10

and 8b+10 = 10+8b base 10

So which is it?

Melody Feb 28, 2020

#4**+1 **

In what bases, b, does (b + 10) divide into (8b + 10) without any remainder?

Well if everything is in base b then we have

b+10=2b (any base)

8b+10=9b (any base)

9b/2b= 4.5

So I am reasonably sure there will be a remainder in any base.

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I suspect the tens in the question are actually meant to be in base 10

in which case guests answer of base 4 does work. Thanks guest for your answer

If b=4 then you have

\(\frac{8b+10}{b+10}_{base\;10}=\frac{8*4+10}{4+10}_{base\;10}=\frac{42}{14}_{base\;10}=3_{Any\;base\;4\; or\; more }\)

Melody
Feb 28, 2020