theachiever wrote:If a,b,c,d and e are positive integers such that ax10^d/bx10^e = cx10^4 ,is bc/a an integer?

1.d-e>=4
2.d-e>4

from the context here, i assume that the stated condition should say a(10^d) / b(10^e) = c(10^4).

the question is asking about the quantity bc/a, so it is most expedient to solve for that quantity.
to do so, multiply both sides by b/a and divide by 10^4, giving bc/a = (10^d) / (10^e)(10^4) = bc/a.
therefore, this problem is asking whether (10^d) / (10^e)(10^4) is an integer.
that expression reduces to 10^(d - e - 4).

note that all powers of 10 from 10^0 upward are integers, and that all negative powers of 10 are non-integers (these are fractions whose denominators are positive powers of 10).
therefore, for the quantity in question to be an integer, the exponent (d - e - 4) must be 0 or more.

for that to be true, d - e must be at least 4. that's guaranteed by either of the statements, so the answer is (d).