SOLUTION: How many positive integers n are there such that 2n+1 is a divisor of 8n+46?

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(52896) (Show Source): You can put this solution on YOUR website!
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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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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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