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A positive integer is called semiprime if it is the product of exactly

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A positive integer is called semiprime if it is the product of exactly [#permalink]

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New post 28 Apr 2017, 01:57
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A positive integer is called semiprime if it is the product of exactly two not-necessarily-distinct prime numbers. A positive integer is called highly composite if it has more factors than any smaller positive integer has. How many positive integers are both semiprime and highly composite?

A. 0
B. 1
C. 2
D. 3
E. Infinitely many
[Reveal] Spoiler: OA

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Re: A positive integer is called semiprime if it is the product of exactly [#permalink]

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New post 28 Apr 2017, 08:17
Examples of Semi-Primes

4 = 2x2
6 = 2x3
10 = 2x5

The smallest Semi Prime number is 4 which has 3 factors. The numbers smaller than 4 are 1,2 & 3. All three have 1,2,& 2 factors respectively. Which means 4 has higher number of factors than the number smiler than it, hence 4 is a semi-prime highly composite number.

The next Semi-Prime is 6 which has 4 factors. The numbers smaller than 5 are 1,2,3,4 & 5. All give have 1,2,2,3 & 2 factors respectively. Which means 6 has higher number of factors than the number smiler than it, hence 6 is a semi-prime highly composite number.

The next Semi-Prime is 10 which has 4 factors. The numbers smaller than 10 do not have factors less than 4. Hence 10 is not a semi-prime highly composite number.

This brings to the point that a semi-prime number if it is a square, will have 3 factors and it the semi-prime number is not a square will have 4 factors. As we can see above that semi-primes after 6 will either have 3 or 4 factors and numbers smaller than them will either than 3 or 4 factors, so semi-primes after 6 will not be a semi prime highly composite. Hence we have only 2 semi prime highly composite: 4 & 6.

Answer is C. 2
Re: A positive integer is called semiprime if it is the product of exactly   [#permalink] 28 Apr 2017, 08:17
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A positive integer is called semiprime if it is the product of exactly

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