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26 May 2017, 04:48
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If the sum of all positive factors of an integer n is 2n, n is a perfect number. For example, the factors of 6 are 1, 2, 3, and 6, and from the sum \(1 + 2 + 3 + 6 = 12 = 2*6\), the sum of the factors of 6 becomes \(12 = 2*6\), thus 6 is the first perfect number. Then, what is the number of factors of the second perfect number?
A. 4
B. 5
C. 6
D. 8
E. 12
A. 4
B. 5
C. 6
D. 8
E. 12
26 May 2017, 04:49
Kudos
Bookmarks
Official Solution:
If the sum of all positive factors of an integer n is 2n, n is a perfect number. For example, the factors of 6 are 1, 2, 3, and 6, and from the sum \(1 + 2 + 3 + 6 = 12 = 2*6\), the sum of the factors of 6 becomes \(12 = 2*6\), thus 6 is the first perfect number. Then, what is the number of factors of the second perfect number?
A. 4
B. 5
C. 6
D. 8
E. 12
In general, perfect number is \((2^{n-1})(2^{n}-1)\) when n=prime number. In other words, when n=2, \((2^{2-1})(2^{2}-1) = (2)(3) = 6\) is the first perfect number. The 2nd perfect number is when n=3, and with substitution, you get \((2^{3-1})(2^{3}-1) = (4)(7) = 28\). In other words, the sum of all factors of \(28 = 1+2+4+7+14+28 = 2(28)\), hence it is a perfect number. From \(28 = (2^{2})(7^{1})\), the number of factors is \((2+1)(1+1) = 6\). The answer is C.
Answer: C
If the sum of all positive factors of an integer n is 2n, n is a perfect number. For example, the factors of 6 are 1, 2, 3, and 6, and from the sum \(1 + 2 + 3 + 6 = 12 = 2*6\), the sum of the factors of 6 becomes \(12 = 2*6\), thus 6 is the first perfect number. Then, what is the number of factors of the second perfect number?
A. 4
B. 5
C. 6
D. 8
E. 12
In general, perfect number is \((2^{n-1})(2^{n}-1)\) when n=prime number. In other words, when n=2, \((2^{2-1})(2^{2}-1) = (2)(3) = 6\) is the first perfect number. The 2nd perfect number is when n=3, and with substitution, you get \((2^{3-1})(2^{3}-1) = (4)(7) = 28\). In other words, the sum of all factors of \(28 = 1+2+4+7+14+28 = 2(28)\), hence it is a perfect number. From \(28 = (2^{2})(7^{1})\), the number of factors is \((2+1)(1+1) = 6\). The answer is C.
Answer: C
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