GreenlightTestPrep wrote:
Positive integer N has k positive divisors, and k has x positive divisors. If k and x are odd integers greater than 2, what is the least possible value of N - k - x?
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IMPORTANT CONCEPT: A perfect square (the square of an integer) will always have an ODD number of positive divisors.
For example, 25 has 3 positive divisors: 1, 5 and 25
Likewise, 16 has 5 positive divisors 1, 2, 4, 8 and 16
And 100 has 9 positive divisors 1, 2, 4, 5, 10, 20, 25, 50 and 100
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Since we're told k is ODD, we know that
N is a perfect square.
Let's test some possible value of N....
case i: It could be the case that N = 4, which means
k = 3 (since the number 4 has
3 positive divisors)
HOWEVER, we're also told x is ODD, which mean
k must also be a perfect square
Since
3 is NOT a perfect square, we must test another value of N
case ii: It could also be the case that N = 9, which means
k = 3 (since the number 9 has
3 positive divisors)
HOWEVER, we're also told x is ODD, which mean
k must also be a perfect square
Since
3 is NOT a perfect square, we must test another value of N
case iii: It could also be the case that N = 16, which means
k = 5 (since the number 16 has
5 positive divisors)
HOWEVER, we're also told x is ODD, which mean
k must also be a perfect square
Since
5 is NOT a perfect square, we must test another value of N
case iv: It could also be the case that N = 25, which means
k = 3 (since the number 25 has
3 positive divisors)
HOWEVER, we're also told x is ODD, which mean
k must also be a perfect square
Since
3 is NOT a perfect square, we must test another value of N
case v: It could also be the case that N = 36, which means
k = 9 (since the number 36 has
9 positive divisors)
In this case,
9 IS a perfect square, which means we've found the SMALLEST value of N that satisfies all conditions.
9 has
3 positive factors, which means x =
3What is the least possible value of N - k - x? Smallest value = 36 -
9 -
3= 24
Answer: D
Cheers,
Brent
_________________
Brent Hanneson - founder of Greenlight Test Prep