oquiella wrote:2. What is the greatest possible length of a positive integer less than 1,000?
Note: For any positive integer n, n>1, the "length" of n is the number of positive primes whose product is n. For example, the length of 50 is 3 since 50= (2) (5) (5)
A. 10
B. 9
C. 8
D. 7
E. 6
To MAXIMIZE the length of n, the prime-factorization of n must include AS MANY PRIME FACTORS AS POSSIBLE.
For this reason, we must MINIMIZE the value of each prime factor.
Since 2 is the smallest prime factor, the prime-factorization of n must include as many 2's as possible.
If n =
2*2*2*2*2*2*2*2*2 = 512, then the prime-factorization of n has nine prime factors, yielding a length of 9.
If the prime-factorization of n includes any more 2's, the value of n will exceed 1000.
Since the prime-factorization n cannot include more 2's than the 9 values in red, the greatest possible length of n = 9.
The correct answer is
B.
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