APICS Mathematics Competitions

 Vezi http://www.math.unb.ca/apics.papers/

APICS Mathematics Contest 1978

 

     Problem 1.  The expression of a positive integer, n in base b is

It is known that the expression of the integer 2n in the same base is

Determine the values of b and n in base 10.

 

ANSWER CiP $b=7$,  $n=480$

      Solution CiP

(1)     $n=1254_{b}=1\cdot b^{3}+2\cdot b^{2}+5\cdot b+4$

$2n=2541_{b}=2\cdot b^{3}+5\cdot b^{2}+4\cdot b+1$

so we have equation

$2(b^{3}+2b^{2}+5b+4)=2b^{3}+5b^{2}+4b+1$

 $\Leftrightarrow b^{2}-6\cdot b-7=0$,

that is, a simple equation of degree 2 whose roots are $b_{1}=-1$ and $b_{2}=7$. But it needs that $b>5$ so $b=7$ and n will be calculated quickly with formula (1), $n=1\cdot 7^{3}+2\cdot7^{2}+5\cdot 7+4=343+98+35+4=480$. It is easily verified that $2n=960=2541_{7}$.

$\blacksquare$