If a, b, c, and d are integers where 0

Question

If a, b, c, and d are integers where 0 < a < b < c < d, what is the value of a + b + c +d?

(1) a, b, c, and d are consecutive integers where (a + b) and (c + d) are prime.

(2) ac + ad + bc + bd = 21

Mohammadmo · Answer

‏I don't see the options here.
But the answer would be 10.
1+2+3+4=10

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Bunuel · Answer

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

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

I suggest you to go through the following posts: 
 ALL YOU NEED FOR QUANT .
 Ultimate GMAT Quantitative Megathread

Hope this helps.

Kinshook · Answer

Given: a, b, c, and d are integers where 0 < a < b < c < d Asked: What is the value of a + b + c +d? (1) a, b, c, and d are consecutive integers where (a + b) and (c + d) are prime. Let us consider few cases. {a,b,c,d} = {1,2,3,4} and 1+2=3 & 3+4=7 are prime YES => a + b + c +d = 10 {a,b,c,d} = {2,3,4,5} and 2+3=5 is prime & 4+5=9 is not prime NO {a,b,c,d} = {3,4,5,6} and 3+4=7 & 5+6=11 are prime YES => a + b + c +d = 18 Since there are 2 cases in which {a,b,c,d} are consecutive integers where (a + b) and (c + d) are prime and give different values of a + b + c +d NOT SUFFICIENT (2) ac + ad + bc + bd = 21 Let us consider few cases. {a,b,c,d} = {1,2,3,4} and ac + ad + bc + bd = 3+4+6+8 = 21 There is no other case possible since {a,b,c,d} are integers where 0 < a < b < c < d Increasing any of the values of {a,b,c,d} will increase the value of ac + ad + bc + bd = 21 SUFFICIENT IMO B

firas92 · Answer

(1) a, b, c, and d are consecutive integers where (a + b) and (c + d) are prime.

If (a,b,c,d)=(1,2,3,4) , a+b+c+d = 10

If (a,b,c,d)=(3,4,5,6) , a+b+c+d = 18

1 is insufficient

(2) ac + ad + bc + bd = 21

a(c+d)+b(c+d)=21
(a+b)(c+d)=21

21=1*21 or 21=3*7

Since we're dealing with positive integers and a<b<c<d, a+b>1

(a+b)(c+d) must be 3*7

Only way this is possible is when (a,b,c,d)=(1,2,3,4)

2 is sufficient

Answer is (B)

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ScottTargetTestPrep · Answer

Solution:

Statement One Alone:

a, b, c, and d are consecutive integers where (a + b) and (c + d) are prime.

We see that b = a + 1, c = a + 2 and d = a + 3 since a, b, c, and d are consecutive integers. Therefore, a + b = 2a + 1 and c + d = 2a + 5. In other words c + d is 4 more than a + b. We are given that (a + b) and (c + d) are primes, however, there are many primes such that one is 4 more than the other. For example, 7 is 4 more than 3 and 17 is 4 more than 13. In the former case, a + b + c + d = 1 + 2 + 3 + 4 = 10, while in the latter, a + b + c + d = 6 + 7 + 8 + 9 = 30. Statement one alone is not sufficient.

Statement Two Alone:

ac + ad + bc + bd = 21

Simplifying, we have:

a(c + d) + b(c + d) = 21

(a + b)(c + d) = 21

Since a, b, c, and d are positive integers, we see that a + b = 3 and c + d = 7 OR a + b = 7 and c + d = 3. Either way, a + b + c + d = 3 + 7 = 7 + 3 = 10. Statement two alone is sufficient.

Answer: B

bumpbot · Answer

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