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If a,b,c, d are distinct even integers and a is prime, is [#permalink]

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06 Jul 2011, 06:22

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E

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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.

Now considering Statement 1

ba is a multiple of 21. So ba should be a multiple of 3 and 7. Factor Foundation rule. 40 is a factor of cd i.e 2^3 * 5. As we do have required 2^3 primes which means that abcd is divisible by 8.

Considering statement 2

bc is divisible by 10 i.e. 2*5. We have one required 2. Considering perfect squares e.g 4 = 2^2 16=2^4

So in any case we have the required three 2s. So this is sufficient as well. So D is the correct answer.

Now can experts on this forum please please tell me that my approach is correct to solve this question?

Re: Distinct even Integers - Please clarify my approach [#permalink]

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06 Jul 2011, 08:27

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Expert's post

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 21

We 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? _________________

GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

Another .... GMAT Terminology Question [#permalink]

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07 Sep 2012, 18:54

I've been stuck at this question for quite sometime. I see the answer. I'm wondering if someone can explain what the answer should be. (and why?)

If a, b, c, and 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 dc. (2) 10 is a divisor of the two digit number bc, and the product of a and d is a perfect square.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Princeton Review (2012-05-22). 1,037 Practice Questions for the New GMAT, 2nd Edition:

Re: Another .... GMAT Terminology Question [#permalink]

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07 Sep 2012, 21:12

SlowGuy wrote:

I've been stuck at this question for quite sometime. I see the answer. I'm wondering if someone can explain what the answer should be. (and why?)

If a, b, c, and 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 dc. (2) 10 is a divisor of the two digit number bc, and the product of a and d is a perfect square.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Princeton Review (2012-05-22). 1,037 Practice Questions for the New GMAT, 2nd Edition:

Lets derive few pointers from this questions.

Firstly, a=2 becasue 2 is only even prime integer. Secondly, we should concentrate on the values for b, c, d becasue if "bcd" is divisible by 8 than the four digit number abcd is divisible by 8.

My answer is E because i'm not able to find out from both the option that whether "bcd" is divisible by 8 or not.

Any takers? _________________

----------------------------------------------------------------------------------------- What you do TODAY is important because you're exchanging a day of your life for it! -----------------------------------------------------------------------------------------

Last edited by Capricorn369 on 08 Sep 2012, 10:47, edited 1 time in total.

Re: Another .... GMAT Terminology Question [#permalink]

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08 Sep 2012, 00:31

Capricorn369 wrote:

SlowGuy wrote:

I've been stuck at this question for quite sometime. I see the answer. I'm wondering if someone can explain what the answer should be. (and why?)

If a, b, c, and 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 dc. (2) 10 is a divisor of the two digit number bc, and the product of a and d is a perfect square.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Princeton Review (2012-05-22). 1,037 Practice Questions for the New GMAT, 2nd Edition:

Lets derive few pointers from this questions.

Firstly, a=2 becasue 2 is only even prime integer. Secondly, we should concentrate on the values for b, c, d becasue if b+c+d is divisible by 8 than the four digit number abcd is divisible by 8.

My answer is E because i'm not able to find out from both the option that whether b+c+d is divisible by 8 or not.

Any takers?

The divisibility rule for 8 says that the three digit number formed by the last digits must be divisible by 8, and not the sum of the last three digits. So, ABC as a three digit number should be divisible by 8 and not the sum A + B + C. See 206 as an example. _________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

Last edited by EvaJager on 08 Sep 2012, 00:56, edited 1 time in total.

Re: Another .... GMAT Terminology Question [#permalink]

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08 Sep 2012, 00:48

2

This post received KUDOS

SlowGuy wrote:

I've been stuck at this question for quite sometime. I see the answer. I'm wondering if someone can explain what the answer should be. (and why?)

If a, b, c, and 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 dc. (2) 10 is a divisor of the two digit number bc, and the product of a and d is a perfect square.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Princeton Review (2012-05-22). 1,037 Practice Questions for the New GMAT, 2nd Edition:

Necessarily \(a=2\) being the only even prime number.

(1) \(ba\) must be an even multiple of \(21,\) and since \(a=2,\) necessarily \(ba = 42,\) so \(b=4.\) \(dc\) must be a multiple of \(40\); the only possibility is \(dc=80\) (\(dc\) cannot be \(40,\) as \(d\) must be different from \(b=4\)). So \(d=8\) and \(c=0.\) In conclusion, \(abcd=2408\). Sufficient. In fact, doesn't matter if this number is divisible by \(8,\) there is just one four digit number which fulfills the given conditions. In particular, \(2408\) is divisible by \(8.\)

(2) \(bc\) is a multiple of \(10,\) therefore \(c=0.\) The product \(2\cdot{d}\) must be a perfect square, and since \(a=2\) and \(d\) must be different from \(a\) and \(c,\) the only possibility is \(d=8.\) The four digit number is \(2b08\), and \(b\) can be either \(4\) or \(6.\) Both \(2408\) and \(2608\) are divisible by \(8.\) Sufficient.

Answer D _________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

Re: Another .... GMAT Terminology Question [#permalink]

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08 Sep 2012, 02:27

Expert's post

SlowGuy wrote:

I've been stuck at this question for quite sometime. I see the answer. I'm wondering if someone can explain what the answer should be. (and why?)

If a, b, c, and 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 dc. (2) 10 is a divisor of the two digit number bc, and the product of a and d is a perfect square.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Princeton Review (2012-05-22). 1,037 Practice Questions for the New GMAT, 2nd Edition:

Merging similar topics. Please refer tot he solutions above.

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