Question 1182795
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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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<pre>
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.      <U>ANSWER</U>
</pre>

Solved.


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