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by theachiever » Fri Dec 21, 2012 10:49 pm
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

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by puneetkhurana2000 » Fri Dec 21, 2012 11:31 pm
Can you please rephrase the question properly?

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by lunarpower » Sat Dec 22, 2012 8:22 pm
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).
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by The Iceman » Mon Dec 24, 2012 4:21 am
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

The problem statement can be reduced to "is 10 ^ (d-e-4) >= 1?" => for a yes we should have (d-e-4) >= 0 => d-e >= 4 and for a no, ofcourse d-e < 4

Now if you see the statements, statement II is a proper subset of the statement I.

Hence, each statement alone is sufficient.
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