document.write( "Question 845227: The product of two positive consecutive odd integers is 1 less than 5 times their sum.
\n" ); document.write( "Find the integers \n" ); document.write( "

Algebra.Com's Answer #509218 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"n-1\"$ = the smallest of the two consecutive odd integers.
\n" ); document.write( " $\"n%2B1\"$ = the next odd integer (the other one of the two consecutive odd integers).
\n" ); document.write( " $\"%28n-1%29%2B%28n%2B1%29=2n\"$ = the sum of the two consecutive odd integers.
\n" ); document.write( " $\"%28n-1%29%2A%28n%2B1%29=n%5E2-1\"$ = the product of the two consecutive odd integers.
\n" ); document.write( "The equation is
\n" ); document.write( " $\"n%5E2-1=5%2A%282n%29-1\"$
\n" ); document.write( "Solving:
\n" ); document.write( " $\"n%5E2-1=5%2A%282n%29-1\"$
\n" ); document.write( " $\"n%5E2-1=10n-1\"$
\n" ); document.write( " $\"n%5E2=10n\"$
\n" ); document.write( "Since $\"n%3C%3E0\"$ , I can divide both sides by $\"n\"$ and get
\n" ); document.write( " $\"n%5E2%2Fn=10n%2Fn\"$
\n" ); document.write( " $\"n=10\"$
\n" ); document.write( "So $\"n-1=10-1\"$ --> $\"n-1=highlight%289%29\"$ , and
\n" ); document.write( " $\"n%2B1=10%2B1\"$ --> $\"n%2B1=highlight%2811%29\"$
\n" ); document.write( "Verification:
\n" ); document.write( "The numbers $\"9\"$ and $\"11\"$ are odd.
\n" ); document.write( "Their sum is $\"9%2B11=20\"$ .
\n" ); document.write( "Five times their sum is $\"5%2A20=100\"$ .
\n" ); document.write( "One less than 5 times their sum is $\"100-1=99\"$ .
\n" ); document.write( "Their product is $\"9%2A11=99\"$ and that is exactly \"1 less than 5 times their sum.\"
\n" ); document.write( "
\n" ); document.write( "NOTES:
\n" ); document.write( "I could have called the two consecutive odd integers $\"x\"$ and $\"x%2B2\"$ ,
\n" ); document.write( "but I thought that would make the calculations more complicated.
\n" ); document.write( "Someone could think to call the two odd integers $\"2k%2B1\"$ and $\"2k%2B3\"$ ,
\n" ); document.write( "or $\"2k-1\"$ and $\"2k%2B1\"$ ,
\n" ); document.write( "but that was going to make it more complicated.
\n" ); document.write( "If there are consecutive odd integers, or consecutive even integers,
\n" ); document.write( "just call them $\"m\"$ and $\"m%2B2\"$ ,
\n" ); document.write( "or $\"m-1\"$ and $\"m%2B1\"$ .
\n" ); document.write( "Those expressions do not say that the numbers are even or odd,
\n" ); document.write( "but they say that they are the same kind (both odd or both even),
\n" ); document.write( "and when you find the solution you can verify if it is the right kind. \n" ); document.write( "

\n" );