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24 Sep 2019, 07:53
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Difficulty:
95%
(hard)
Question Stats:
49% (03:04) correct
51%
(02:41)
wrong
based on 177
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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?
A) -2
B) 9
C) 17
D) 24
E) 36
A) -2
B) 9
C) 17
D) 24
E) 36
14 Oct 2019, 10:24
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----ASIDE--------------------
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
-------------------------
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 = 3
What is the least possible value of N - k - x?
Smallest value = 36 - 9 - 3
= 24
Answer: D
Cheers,
Brent
General Discussion
24 Sep 2019, 08:26
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Odd integers means all are SQUARES..
We will have to take least value of N, for that k has to be least and in turn x has to be least
k has x positive divisors, so least value of x can be 3...
3 positive divisors means perfect square of a prime number, so 3^2=9 as 3 is the smallest odd prime number
Now N has 9 prime factors..
Different values of N ..
1) 9=1*9=(8+1) or \(a^8\)...Minimum value 2^8=256
2) 9=3*3=(2+1)(2+1) or \(a^2b^2\).. Taking least value of a and b, we get \(2^23^2=36\)
Thus N-k-x=36-9-3=24
D
