Bunuel wrote:
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
The question is based on factorisation.
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2010/1 ... ly-number/https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2010/1 ... t-squares/Semi-prime numbers have exactly 2 prime factors - same or different.
So a semi prime number N, would be a^2 or ab where a and b are both primes.
So N would have 3 or 4 factors. For it to also be highly composite, all positive integers (starting from 1) less than N should have 3 or 4 factors. The moment you get 5 factors or more, you cannot have a number which is both semi prime and highly composite.
The first number to have 5 factors will be 2^4 = 16
It will easier to get 6 factors with 2^2 * 3 = 12
So we just need to check numbers less than 12.
4 = 2*2 has 3 factors... It is semi prime and highly composite because 3, 2 and 1 have fewer factors.
6 = 2*3 has 4 factors... It is semi prime and highly composite because 5, 4, 3, 2 and 1 have fewer factors.
Rest of the numbers smaller than 12 will have 2 or 3 or 4 factors so none of them can be highly composite.
Answer (C)
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