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E
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Difficulty:
95%
(hard)
Question Stats:
41% (02:41) correct
59%
(02:48)
wrong
based on 140
sessions
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Date
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For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.
If k is a positive integer such that \(k^2\) is divisible by 45 and 80, what is the units digit of \([\frac{k^3}{4000}]\)?
A. 0
B. 1
C. 7
D. 4
E. Can not be determined.
If k is a positive integer such that \(k^2\) is divisible by 45 and 80, what is the units digit of \([\frac{k^3}{4000}]\)?
A. 0
B. 1
C. 7
D. 4
E. Can not be determined.
![For any positive number x, the function [x] denotes the greatest integ](/forum/styles/gmatclub_light/imageset/icon_post_target_unread.svg "For any positive number x, the function [x] denotes the greatest integ"){kind=icon}
26 Sep 2018, 08:22
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GmatDaddy
Please correct your choices..
We are looking for units digit and choice C and D give you 2-digit integers.
Anyways..
k^2 divisible by 45 or 3^2*5, so k will surely be a multiple of 3*5
k^2 is also divisible by 80 or 2^4*5 or 2^3*2*5 so k is surely a multiple of 2*2*5
Therefore k will surely be a multiple of LCM of 3*5 and 2*2*5 or 2*2*3*5 or 60
Therefore k^3 = (60a)^3=216000a^3
Therefore k^3/4000=216000a^3/4000=54a^3
Now units digit will depend on a..
Say a is 1 Ans is 4
a is 7, Ans is 4*3 or 2
So cannot be determined
E
General Discussion
29 Sep 2018, 17:55
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GmatDaddy
If a number is divisible by 45 and 80, then it’s divisible by the least common multiple of 45 and 80, which is 720. The prime factorization of 720 is 9 x 8 x 10 = 3^2 x 2^3 x 2 x 5 = 2^4 x 3^2 x 5. Therefore, k^2 is divisible by 2^4 x 3^2 x 5 and k must be divisible by 2^2 x 3 x 5 = 60. In other words, k is a multiple of 60.
If k = 60, then k^3/4000 = 60^3/4000 = (60 x 60 x 60)/4000 = (6 x 6 x 6)/4 = 54. So the units digit of [k^3/4000] is 4.
If k = 600, then k^3/4000 = 600^3/4000 = (600 x 600 x 600)/4000 = (60 x 60 x 60)/4 = 54,000. So the units digit of [k^3/4000] is 0.
We see that the units digit of [k^3/4000] is not unique. So it can’t be determined.
Answer: E
