chrisdkoning wrote:(30^30) × (29^29) × (28^28) × . . . × (3^3) × (2^2) × (1^1) = N
What is the highest value of K, such that N/(125^K) is an integer?
How do I solve this, and is this question a GMAT level problem? Any help would be appreciated!
(30^30) × (29^29) × (28^28) × . . . × (3^3) × (2^2) × (1^1).
125 = 5³.
To determine how many times 5³ can divide into N, count how many 5's are contained within the product above.
30³� = 5³�6³� --> thirty 5's.
25²� = (5²)²� = 5�� --> fifty 5's.
20²� = (5²�)(4²�) --> twenty 5's.
15¹� = 5¹�3¹� --> fifteen 5's.
10¹� = 5¹�2¹� --> ten 5's.
5� = five 5's.
Total number of 5's = 30+50+20+15+10+5 = 130.
Implication:
N = (5¹³�)(other factors).
Thus:
N/(125^k) =
[(5¹³�)(other factors)]/
[(5³)^k].
The values in blue reveal that the greatest possible value for k is
43, as follows:
[
(5¹³�)(other factors)]/
[(5³)�³]
= [
(5¹³�)(other factors)]/
5¹²�
=
(5)(other factors).
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