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?
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.
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