Basic Bases

In base b b , 2 4 b 2 = 52 1 b 24_{b}^{2}=521_{b} . Find b b .


Question from the Australian Intermediate Maths Olympiad 2014.


The answer is 15.

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1 solution

Sudeep Salgia
Nov 30, 2015

2 4 b 2 = 52 1 b ( 2 b + 4 ) 2 = ( 5 b 2 + 2 b + 1 ) \displaystyle 24_b^2 = 521_b \Rightarrow (2b + 4)^2 = (5b^2 + 2b + 1)
4 b 2 + 16 b + 16 = 5 b 2 + 2 b + 1 b 2 14 b 15 = 0 ( b 15 ) ( b + 1 ) = 0 \displaystyle \Rightarrow 4b^2 + 16b + 16 = 5b^2 + 2b + 1 \Rightarrow b^2 -14b -15 = 0 \Rightarrow (b-15)(b+1) = 0
b = 15 \displaystyle \Rightarrow b = 15 or b = 1 b = -1 . Since b > 0 b > 0 , b = 15 \displaystyle \boxed{b=15}

Well done!

A Former Brilliant Member - 5 years, 6 months ago

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