document.write( "Question 1158010: When the number 𝑛 is written in base 𝑏, its representation is the two-digit number 𝐴𝐵 where 𝐴=𝑏−2 and 𝐵=2. What is the representation for 𝑛 in base (𝑏−1)? \n" ); document.write( "

Algebra.Com's Answer #854539 by KMST(5377) $\"\"$ $\"About$

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The value of a number in any base is really a polynomial.
\n" ); document.write( "If ABCD is how a number is written in base $\"x\"$ , its value is
\n" ); document.write( " $\"Ax%5E3%2BBx%5E2%2BCx%2BD\"$ .
\n" ); document.write( "Of course, if ABCD is how a number is written in base $\"x\"$ ,
\n" ); document.write( "we know that $\"x\"$ is an integer,
\n" ); document.write( "that $\"x\"$ is greater than A, B, C, and D,
\n" ); document.write( "and that $\"x\"$ would be written as 10 in base $\"x\"$
\n" ); document.write( "The value of number 324 in base 10 is $\"3%2A10%5E2%2B2%2A10%2B4\"$ .
\n" ); document.write( "
\n" ); document.write( "The value of the number $\"n\"$ written as $\"AB\"$ in base $\"b\"$ is
\n" ); document.write( " $\"n=Ab%2BB\"$ , and we know that $\"b%3EA=b-2\"$ and $\"b%3EB=2\"$ .
\n" ); document.write( "So, $\"b%3E=3\"$ and $\"b-1%3E=2\"$ .
\n" ); document.write( "Substituting $\"b-2\"$ for $\"A\"$ and $\"2\"$ for $\"B\"$ , we get
\n" ); document.write( " $\"n=%28b-2%29b%2B2\"$
\n" ); document.write( "Now we are dealing with polynomials.
\n" ); document.write( " $\"n=%28b-2%29b%2B2\"$ --> $\"n=b%5E2-2b%2B2\"$ --> $\"n=b%5E2-2b%2B1%2B1\"$ --> $\"n=%28b-1%29%5E2%2B1\"$ --> $\"Highlight%28n=1%2A%28b-1%29%5E2%2B0%2A%28b-1%29%2B1%29\"$
\n" ); document.write( "and that polynomial in $\"%28b-1%29\"$ is the value of the number that in base $\"b-1\"$ would be written as $\"highlight%28101%29\"$ . \n" ); document.write( "

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