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16 Sep 2014, 01:45
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The product of two negative integers, \(a\) and \(b\), is a prime number \(p\). If \(p\) is the number of positive factors of a positive integer \(n\), where \(n\) is NOT a perfect square, what is the value of the median of the four integers \(a\), \(b\), \(p\), and \(n\)?
A. \(0\)
B. \(\frac{1}{2}\)
C. \(1\)
D. \(\frac{3}{2}\)
E. \(2\)
A. \(0\)
B. \(\frac{1}{2}\)
C. \(1\)
D. \(\frac{3}{2}\)
E. \(2\)
16 Sep 2014, 01:45
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Official Solution:
The product of two negative integers, \(a\) and \(b\), is a prime number \(p\). If \(p\) is the number of positive factors of a positive integer \(n\), where \(n\) is NOT a perfect square, what is the value of the median of the four integers \(a\), \(b\), \(p\), and \(n\)?
A. \(0\)
B. \(\frac{1}{2}\)
C. \(1\)
D. \(\frac{3}{2}\)
E. \(2\)
This question covers several concepts in number theory and is quite challenging.
Let's begin with the positive integer \(n\), which we know is not a perfect square. We can recall that positive perfect squares have an odd number of factors while all other positive integers have an even number of factors. Hence, the number of factors of \(n\), which is \(p\), must be even. We are also told that \(p\) is a prime number, and since the only even prime number is \(2\), it follows that \(p=2\). Moreover, since only prime numbers have exactly \(2\) factors (1 and itself), \(n\) must also be a prime number.
Given that \(ab=p=2\), we have either \(a=-1\) and \(b=-2\), or vice versa. Therefore, the set of four integers is {-2, -1, 2, some prime}. The median of this set is \(\frac{-1+2}{2}=\frac{1}{2}\).
Answer: B
The product of two negative integers, \(a\) and \(b\), is a prime number \(p\). If \(p\) is the number of positive factors of a positive integer \(n\), where \(n\) is NOT a perfect square, what is the value of the median of the four integers \(a\), \(b\), \(p\), and \(n\)?
A. \(0\)
B. \(\frac{1}{2}\)
C. \(1\)
D. \(\frac{3}{2}\)
E. \(2\)
This question covers several concepts in number theory and is quite challenging.
Let's begin with the positive integer \(n\), which we know is not a perfect square. We can recall that positive perfect squares have an odd number of factors while all other positive integers have an even number of factors. Hence, the number of factors of \(n\), which is \(p\), must be even. We are also told that \(p\) is a prime number, and since the only even prime number is \(2\), it follows that \(p=2\). Moreover, since only prime numbers have exactly \(2\) factors (1 and itself), \(n\) must also be a prime number.
Given that \(ab=p=2\), we have either \(a=-1\) and \(b=-2\), or vice versa. Therefore, the set of four integers is {-2, -1, 2, some prime}. The median of this set is \(\frac{-1+2}{2}=\frac{1}{2}\).
Answer: B
05 Aug 2016, 04:30
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nahid78
Yes.
Tips about the perfect square:
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;
3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);
4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.
Hope it helps.
