Olympiad workbookIOQMBinary Numbers

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 ?

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aligned & N=3 b^0+0 b^1+5 b^2 & =5 (b+2)^0+0 (b+2)^1+3(b+2)^2 & N=5 b^2+3=3(b+2)^2+5=3 (b^2+4 b+4 )+5 & 2 b^2-12 b-14=0 & b^2-6 b-7=0 & (b+1)(b-7)=0 & b=7(b=-1 rejected ) & N=5 7^2+3=248 ; Product of digits =64 aligned

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