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# Bases Question

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

Feb 27, 2020

#1
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b + 10 divides 8b + 10 for b = 4.

Feb 27, 2020
#2
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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?

Feb 28, 2020
#3
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Base B

#4
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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
edited by Melody  Feb 28, 2020