enigma123 wrote:
Question is:
If a,b,c, d are distinct even integers and a is prime, is the four digit number abcd divisible by 8?
1) The two digit number ba is a multiple of 21 and 40 is a factor of the two digit number cd.
2) 10 is a divisor of the two digit number bc and the product of ad is a perfect square.
Here is my approach to solve this mathematically. I have used the Prime Factor approach.
Considering The Stem of the Question:
a=2 and b,c,d could be any even integers. So for abcd to be divisiblle by 8 the abcd should have at least 2^3 prime factors.
I think you're answering a different question from what is intended. With your approach, you're determining whether the product a*b*c*d will be divisible by 8 (you're counting how many 2s are prime factors of a, b, c and d). In fact, from the stem alone, the product a*b*c*d must be divisible by 2^4 at least, since each of the letters is even.
Instead the question means to say (and they aren't sufficiently clear about this) that a, b, c and d are digits in the four-digit number abcd. That is, a is the thousands digit, b the hundreds digit, c the tens digit and d the units digit. We need to know whether this number is divisible by 8. A number is divisible by 8 if its last 3 digits form a number divisible by 8, so we need to know if the 3-digit number bcd is divisible by 8.
This now becomes one of those "logic with digits" problems that really aren't very common on the GMAT, though I suppose they appear on occasion. This type of question is not worth spending a huge amount of time on, however. Still, we can solve:
Now, what do we know? From the stem we know a, b, c and d are *distinct* even digits, and that a = 2, so b, c, and d, the digits we actually care about, must be different values taken from 0, 4, 6 and 8. Now from Statement 1 we know:
The two digit number ba is a multiple of 21We know that a=2, so the two-digit number b2 is a multiple of 21, and b must be 4.
and 40 is a factor of the two digit number cd.If 40 is a factor of cd, then 10 is clearly a factor of cd, and d=0. So c=4 or c=8. But we know that b=4, and c cannot be the same as b, so c=8.
Since we know now that our number abcd is exactly equal to 2480, of course we can answer any divisibility question about our number, so the statement is sufficient.
From Statement 2, we know that
10 is a divisor of the two digit number bc in which case it must be that c=0, since multiples of 10 end in 0.
and the product of ad is a perfect square.Further, the product ad, which is equal to 2d, is a perfect square. Since d can only be 4, 6 or 8, (we've used 0 already), we can see that d must be 8. So we know that our number is 2b08, where b is either 4 or 6, and since 408 and 608 are each divisible by 8, we have sufficient information.
Now, there's either a typo in the original post or in the book, because the statements contradict each other. In Statement 1, perhaps it says 'dc' instead of 'cd' in the original source?
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