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Recall that an integer d is said to be a divisor of an integer a if a/d is also an integer. For how many integers a between -200 and -1 inclusive is the product of the divisors of a negative?

 Aug 25, 2023
 #1
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The product of the divisors of a positive integer is positive, and the product of the divisors of a negative integer is negative. Therefore, the integers a between -200 and -1 inclusive for which the product of the divisors is negative are precisely the odd integers in this range. There are 2−200−(−1)+1​=100 odd integers in this range, so the answer is 100​.

 Aug 25, 2023
 #2
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A non-square  integer always has an even number of  divisors....so...if we  consider the negative  divisors of the  integers between -200 and -1  inclusive.....the product of an even  number of negative  divisors will be positive

 

A perfect square always has an odd number of divisors

So...if we  consider  the negatives to contain "perfect squares"....the integers -1, -4, -9, -16, -25, -36, -49, -64, -81, -100,-121, -144,-169,-196     will all  have an  odd  number  of negative divisors....and the  product of an odd number of negatives will always be  a negative

 

So, 14 integers

 

cool cool cool

 Aug 25, 2023
 #3
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thanks!

Guest Aug 25, 2023

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