niketdoshi123 wrote:The "length of integer x" refers to the number of prime factors, not necessarily distinct, that x has. (If x = 60, the length of x would be 4 because 60 = 2 × 2 × 3 × 5.) What is the greatest possible length of integer z if z < 1,080?
a)12
b)10
c)9
d)7
e)6
This is a MAX/MIN problem.
To MAXIMIZE the length of z -- in other words, to maximize the NUMBER of prime factors in z's prime-factorization -- we need to MINIMIZE the size of each prime factor.
The smallest prime factor is 2.
Thus, to maximize the length of z, we want its prime-factorization to include as many 2's as possible.
Every test-taker should memorize the powers of 2 up to 2^10.
Let z = 2^10 = 1024.
Since 1024 has 10 prime factors -- (2*2*2*2*2)*(2*2*2*2*2) -- it has a length of 10.
For z to have a length of 11, its prime-factorization would need to include an 11th prime factor.
But the smallest prime factor that could be added is 2, and z = 2^11 = 2048 is too great, since z<1080.
Thus, the maximum possible length of z is 10.
The correct answer is
B.
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