guerrero wrote:Hello All.
Please help me understand the approach to solve the problem. I am finding it so tough to understand
g(x) is defined as the product of all even integers k such that 0<k
≤x. For example, g(
14)=2*4*6*8*10*12*14. If g(y) is divisible by 4^11, what is the smallest possible value of Y?
a)22
b)24
c)28
d)32
e)44
thanks in advance !
The problem seems to have been transcribed incorrectly.
The changes in red (see above) reflect what I believe is intended.
4^11 = (2²)^11 = 2^22.
For g(y) to be divisible by 2^22, its prime-factorization must include twenty-two 2's.
We can plug in the answers, which represent the smallest possible value of y.
To determine the number of 2's included in the prime-factorization of g(y), count how many times EACH POWER OF 2 can be divided into g(y).
Answer choice A: 22
g(22) = 2*4.....20*22.
In the prime-factorization of g(22):
Every multiple of 2 provides at least one 2:.
22/2¹ = 11.
The calculation above indicates that g(22) includes 11 multiples of 2¹.
Every multiple of 2² adds a second 2:
22/2² = 22/4 = 5.
The calculation above indicates that g(22) will include five multiples of 2², increasing the total number of 2's by 5.
Every multiple of 2³ adds a third 2:
22/2³ = 22/8 = 2.
The calculation above indicates that g(22) will include two multiples of 2³, increasing the total number of 2's by 2.
Every multiple of 2^4 adds a fourth 2:
22/2^4 = 20/16 = 1.
The calculation above indicates that g(22) will include one multiple of 2^4, increasing the total number of 2's by 1.
Thus, the total number of 2's that can be divided into g(22) = 11+5+2+1 = 19.
Three more 2's are needed.
Thus, the prime-factorization must be expanded to include 24=2*2*2*3, implying that the smallest possible value of y is 24.
The correct answer is
B.
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