Question 1182795: How many positive integers n are there such that 2n+1 is a divisor of 8n+46?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52890)   (Show Source):
You can put this solution on YOUR website! .
How many positive integers n are there such that 2n+1 is a divisor of 8n+46?
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Let's assume that (2n+1) is a divisor of N = 8n+46.
Notice that that (2n+1) is a divisor of the number M = 8n+4 (simply because 8n+4 = 4*(2n+1) ).
It implies that the difference N-M is a multiple of (2n+1), too.
But the difference N-M is equal to (8n+46) - (8n+4) = 46-4 = 42.
Thus the number (2n+1) is a divisor of the number 42.
We can list all the ODD divisors of the number 42.
They are 3, 7 and 21, giving these equations to determine n
2n+1 = 3; 2n+1 = 7 and 2n+1 = 21.
These equations have the following solutions, respectively
n = 1; n = 3 and n = 10.
So, we solved the problem and found out all opportunities for n as 1, 3 and 10. ANSWER
Solved.
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Happy learning, enjoy and be happy (!)
Answer by greenestamps(13209)  (Show Source):
You can put this solution on YOUR website!
By long division or any other method, determine that
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.
On the right side of the equation, 4 is also an integer; and that means 42/(2n+1) must be an integer.
So (2n+1) is a divisor of 42; and since n is an integer, (2n+1) is odd.
The only odd divisors of 42 are 3, 7, and 21.
2n+1=3 --> n=1
2n+1=7 --> n=3
2n+1=21 --> n=10
ANSWER: There are exactly three integers -- 1, 3, and 10 -- for which 2n+1 is a divisor of 8n+46.
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