lisannemuns wrote:Hello, Can somebody help me with this question?
If [p] is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for wich 3^k is a factor of [p]?
a)10
b)12
c) 14
d) 16
e) 18
[p] = 30!.
3^k = the number of 3's that can be divided into [p].
A very fast approach is to count how many times EACH POWER OF 3 can be divided into 30:
30/3¹ = 10.
The calculation above indicates that 30! includes 10 multiples of 3¹.
30/3² = 30/9 = 3.
The calculation above indicates that 30! includes 3 multiples of 3².
30/3³ = 30/27 = 1.
The calculation above indicates that 30! includes 1 multiple of 3³.
Thus, [p] includes 10 multiples of 3, 3 multiples of 3², and 1 multiple of 3³.
Thus, the total number of 3's that can be divided into [p] = 10+3+1 = 14.
The correct answer is
C.
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