document.write( "Question 1182795: How many positive integers n are there such that 2n+1 is a divisor of 8n+46? \n" ); document.write( "

Algebra.Com's Answer #812878 by greenestamps(13215) $\"\"$ $\"About$

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\n" ); document.write( "By long division or any other method, determine that

\n" ); document.write( " $\"%288n%2B46%29%2F%282n%2B1%29=4%2B42%2F%282n%2B1%29\"$

\n" ); document.write( "If 2n+1 is to be a divisor of 8n+46, then the left side of the equation (8n+46)/(2n+1) is an integer.

\n" ); document.write( "On the right side of the equation, 4 is also an integer; and that means 42/(2n+1) must be an integer.

\n" ); document.write( "So (2n+1) is a divisor of 42; and since n is an integer, (2n+1) is odd.

\n" ); document.write( "The only odd divisors of 42 are 3, 7, and 21.
\n" ); document.write( "2n+1=3 --> n=1
\n" ); document.write( "2n+1=7 --> n=3
\n" ); document.write( "2n+1=21 --> n=10

\n" ); document.write( "ANSWER: There are exactly three integers -- 1, 3, and 10 -- for which 2n+1 is a divisor of 8n+46.

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