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Help!

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If $$554_b$$ is the base b representation of the square of the number whose base b representation is $$24_b$$ then find b.

Jan 5, 2019

#1
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$$554_b = (24_b)^2\\ 5b^2+5b+4 = (2b+4)^2 = 4b^2+16b+16\\ b^2- 11b-12=0\\ (b-12)(b+1) = 0\\ b=12,~-1\\ \text{-1 is not a base}\\ b=12$$

$$554 _{12} = 4+60+720 = 784_d\\ 24_{12}=24+4=28_d\\ 748=28^2$$

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Jan 5, 2019