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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Binary Numbers

Olympiad workbook · IOQM · Binary Numbers

A number N in base 10 , is 503 in base b and 305 in base b+2 . What...

A number

N

in base 10 , is 503 in base

b

and 305 in base

b+2

. What is the product of the digits of

N

?

Correct answer

64

Step-by-step solution

\begin{aligned} & N=3 \times b^0+0 \times b^1+5 \times b^2 \\ & \quad=5 \times(b+2)^0+0 \times(b+2)^1+3(b+2)^2 \\ & N=5 b^2+3=3(b+2)^2+5=3\left(b^2+4 b+4\right)+5 \\ & \Rightarrow 2 b^2-12 b-14=0 \\ & \Rightarrow b^2-6 b-7=0 \\ & \Rightarrow(b+1)(b-7)=0 \\ & b=7(b=-1 \text { rejected }) \\ & N=5 \times 7^2+3=248 ; \text { Product of digits }=64 \end{aligned}

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