Is √abcd an integer, where a,b,c,d are all prime numbers?

Question

(1) a+b=c+d
(2) c=2

rajarshee · Answer

for √abcd to be an integer you need at least 2 pairs of same prime numbers.

1. a+b=c+d -> well good but doesn't let us conclude if a=c or a=d as 5+19 = 7+17. all 4 are prime yet different numbers. so insufficient
2. c=2 doesn't give us anything about a,b or d.

both together -> note that addition of 2 prime numbers = even (odd+odd) except when one of the prime numbers is 2. 2 is the only even prime number. so that makes c+d either odd or c+d=4 (when both are 2).
if c+d=odd then a+b=odd then either a or b = 2 and the other number is equal to d. we got 2 pairs of same prime numbers. so integer.

if c+d=4 then of course a,b,c,d = 2 which is also sufficient.

So option C

Dabhishek · Answer

i didnt really get the explaination. will you plz explain in some other way? by \sqrt{abcd} what i get is... multiplication must be perfect square to get integer. what to do next? :roll:

rajarshee · Answer

for √abcd to be an integer abcd has to be the square of an integer. now as all are prime numbers they cannot be all different primes. either all of them are the same prime number (notice it has not been said that they are different primes) or we have 2 pairs of primes. (such as a=c and b=d -> so abcd = a*a*b*b

so all we have to identify from 1 & 2 is if that's true or not. the rest of the solution is in my post earlier. hope this helps.

Dabhishek · Answer

ya got it....thanks bro

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is √abcd an integer, where a,b,c,d are all prime numbers?

(1) a+b=c+d
(2) c=2

There are 4 variables (a,b,c,d) and we need 4 equations in order to solve the question, but there are only 2 equations given from the 2 conditions so (C) has a high chance of being our answer.
Looking at the conditions together,
If c=2=b=a=d, the answer to the question becomes 'yes' as sqrt (abcd)=4.
If c=2 and d is a prime number excluding 2, a=2, b=d, or b=2, a=d, so the answer to the question becomes 'yes' as sqrt (abcd)=sqrt(2^2d^2)=2d. This is also a sufficient condition.
However, this is also one of the common mistakes, so if we apply the Common Mistake Type 4(A), the question is answered 'no' when a=3, b=11,c=d=7, and 'yes' when a=b=c=d=2 in condition 1. This is insufficient.
For condition 2, we cannot know the values of a,b,d, so this is insufficient so the answer becomes (C).

For cases where we need 3 more equation, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

whoisthere · Answer

what's the source of this problem? Is this really GMAT type?

robu · Answer

When a ,b ,c d are integer, does it not mean that all shall be different. Can we take All variable be same i.e. 2

Bunuel · Answer

Unless otherwise specified, different variables can represent the same number.

bumpbot · Answer

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Is √abcd an integer, where a,b,c,d are all prime numbers?

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Is √abcd an integer, where a,b,c,d are all prime numbers?

Prep Toolkit

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My Rewards