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 k ^ 3 / 4000?
Answers:  A) 0  B) 1  C) 27  D) 54  E) can not be determined
<gadget id="calculator">3 * 2</gadget>
<output>6</output>

<gadget id="calculator">6 * 6</gadget>
<output>36</output>

<gadget id="calculator">36 * 6</gadget>
<output>216</output>

<gadget id="calculator">216 / 4</gadget>
<output>54</output>

<result>D</result>
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 k ^ 3 / 4000?
divide(multiply(multiply(multiply(3, 2), multiply(3, 2)), multiply(3, 2)), const_4)
"factorization of 45 = 3 * 3 * 5 factorization of 80 = 5 * 2 ^ 4 means the smallest value of k is \ sqrt { 3 ^ 2 * 5 * 2 ^ 4 } = 3 * 5 * 2 ^ 2 k ^ 3 = 3 ^ 3 * 5 ^ 3 * 2 ^ 6 * x = 3 ^ 3 * 2 * 4000 * x where x can be any integer k ^ 3 / 4000 = 3 ^ 3 * 2 * x = 54 x which obviously has different units digit depending on x answer : d"