If a is an integer and (a^2)/(12^3) is odd, which of the following must be an odd integer? Choose the correct option
A) a/4  B) a/12  C) a/27  D) a/36  E) a/72
we can rewrite: a^2 / 2^6 x 3^3   so a^2 must be at least 2^6*3^4   or a is at least 2^3 * 3^2   A. if we divide by 4, or 2^2, we still have a factor of 2 left out, which will make an even integer..so no   B. if we divide by 12 (3x2^2), we are still left with a factor of 2- so no   C. 27=3^3. it might be even a non-integer, so no. moreover, we are left with 2^3 ..which will make the number even.   D. 36 = 2^2 * 3^2. still we have at least one factor of 2, so no.   E. by poe, E is left..72=2^3 * 3^2. since we know that a is at least this number (other variations would include another odd factor)..then dividing it by this will always yield an odd integer.   ANSWa/72R:a/72
<result>E</result>
If a is an integer and (a^2)/(12^3) is odd, which of the following must be an odd integer?