Find the number of positive integers n such 2 0 n + 2 can divide 2 0 0 3 n + 2 0 0 2 .
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Nice but I feel like I cheated a bit by using Python to find 1 8 0 1 7 = 4 3 × 4 1 9 .
By the way, isn't it enough to show that there is no integer n such that 2 0 n + 2 divides 1 8 0 1 . 7 because 1801.7 is not a integer while 2 0 n + 2 is?
Let Q = 2 0 n + 2 2 0 0 3 n + 2 0 0 0 2 = 2 0 n + 2 1 0 0 ( 2 0 n + 2 ) + 1 8 0 2 = 1 0 0 + 2 0 n + 2 1 8 0 2 . For Q to be a positive integer 2 0 n + 2 1 8 0 2 = 2 0 n + 2 2 × 1 7 × 5 3 = 2 × ( 1 0 n + 1 ) 2 × 1 7 × 5 3 = 1 0 n + 1 1 7 × 5 3 must be a positive integer.
It is simple to notice that there is no value for n which would satisfy the conditions, hence the answer is 0 .
Disclaimer: This question seems similar to my question
A simple question which can be solved by polynomial division and knowing the fact that if number is divided by its factor the remainder is zero.The given dividend is 2003n +2002 and divisor is 20n+2. On dividing the dividend by divisor the remainder obtained is 3n+1802. Since the question asks us to find such a "n" such that the divisor is factor of the dividend, which implies the remainder = 0, 3n + 1802 =0, => n= -600.66667 . But the question asks for the number of positive integers and our answer is negative as well as not an integer. Hence, clearly the answer is zero.....
-266 doesn't satisfy. I think the only integral value is -42.
How did you get the remainder as (3n+798) ???
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Good point out. Somehow by faulty subtraction I got 798 as answer. Thankyou very much for pointing it out. But it did not affect our answer n is still negative while the question asks for positive integers... I have edited the numbers in my answer.
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We want to find the number of positive integers n such that 2 0 n + 2 2 0 0 3 n + 2 0 0 2 = m ∈ Z Simple polynomial division gives: m = 2 0 n + 2 2 0 0 3 n + 2 0 0 2 ⟹ 2 0 m = 2 0 2 0 0 3 + 2 0 n + 2 1 8 0 1 . 7 = 2 0 0 3 + 1 0 n + 1 1 8 0 1 7 On factorizing, 1 8 0 1 7 = 4 3 × 4 1 9 .Therefore, it's only factors are 1 , 4 3 , 4 1 9 , 1 8 0 1 7 .None of these is of the form 1 0 n + 1 .Hence there is no positive integer n which will make 2 0 n + 2 2 0 0 3 n + 2 0 0 2 and consequentially m an integer.