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If i, a and b are integers, is 4(3b + 2) = 5a?  [#permalink]

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If i, a and b are integers, is 4(3b + 2) = 5a?

(1) If i is divided by 5 the quotient is a and the remainder is 3
(2) If i is divided by 12 the quotient is b and the remainder is 11

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Originally posted by reto on 17 May 2015, 10:00.
Last edited by reto on 23 May 2015, 02:25, edited 2 times in total.
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Re: If i, a and b are integers, is 4(3b + 2) = 5a?  [#permalink]

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Cool question, and somewhat abstract. My first inclination on this problem was to think about primes, but then I realized that the problem is actually simpler than that. Distributing out the 4 to reorganize the given question stem, we have the question: "is 12b + 8 = 5a? " Statement 1 shows us that if we multiply 5*a, we will end up with "i - 3," because when "i" is divided by 5 you end up with a and then 3 more (remainder 3). The same concept applies to statement 2, where we see that 12*b will give us "i - 11." So substituting these expressions back into the initial question, we have "Is i - 11 + 8 = i - 3?" It then becomes apparent that they are equal, because "i-3 = i-3."

This entire problem revolved around putting everything in terms of "i." The biggest hint to do this was the fact that "i" was not even present in the initial questioned formula, so we had to incorporate it into the problem.

I hope this helps!
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Re: If i, a and b are integers, is 4(3b + 2) = 5a?  [#permalink]

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We have to just find if 4(3b+2)=5a or not.

Statement 1: i=5a+3
No info. about b to prove the relation to be true.
Insufficient.

Statement 2: i=12b+11
Same as st.1
Insufficient.

Both statements together give us

5a+3=12b+11
5a=4(3b+2)

Hence C
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If i, a and b are integers, is 4(3b + 2) = 5a?  [#permalink]

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reto wrote:
If i, a and b are integers, is 4(3b + 2) = 5a?

(1) If i is divided by 5 the quotient is a and the remainder is 3
(2) If i is divided by 12 the quotient is b and the remainder is 11

My solution is

Statement 1 : i = (5a + 3) .

now a could be 1 , 2 , 3 ,4 and so on .

now if a would be 1 , 5a would be 5
a would be 2 , 5a would be 10 .

Therefore 5a of the equation 4(3b+2) = 5a could be odd or even , but the left hand side 4( 3b +2 ) would be always even .
Hence the equation could be equal in case a =2 , could not be equal (sure shot) when a = 1 .

Statement 2 : i = 12 b + 11

b could be 1,2,3,4 you can add 0 also

but it does not matter what is the value of b is as long as we dont know the value of a , the statement is insufficient .

combined together , we will use algebraic approach to prove their equality .

12 b + 8 = 5a

i-11 + 8 = i -3

i-3 = i-3 , Hence the solution is C
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Re: If i, a and b are integers, is 4(3b + 2) = 5a?  [#permalink]

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VeritasPrepBrandon wrote:
Cool question, and somewhat abstract. My first inclination on this problem was to think about primes, but then I realized that the problem is actually simpler than that. Distributing out the 4 to reorganize the given question stem, we have the question: "is 12b + 8 = 5a? " Statement 1 shows us that if we multiply 5*a, we will end up with "i - 3," because when "i" is divided by 5 you end up with a and then 3 more (remainder 3). The same concept applies to statement 2, where we see that 12*b will give us "i - 11." So substituting these expressions back into the initial question, we have "Is i - 11 + 8 = i - 3?" It then becomes apparent that they are equal, because "i-3 = i-3."

This entire problem revolved around putting everything in terms of "i." The biggest hint to do this was the fact that "i" was not even present in the initial questioned formula, so we had to incorporate it into the problem.

I hope this helps!

Pl see if my approach is correct, VeritasPrepBrandon

We have to prove, if 4(3b+2) = 5a
(1) says, i=5a+3, we don’t know the value of b. INSUFFICIENT
(2) says, i=12b+11, we don’t know the value of a. INSUFFICIENT
Now, (1) + (2), 12b+11 = 5a+3 => 12b+8=5a => 4(3b+2) = 5a. Option C correct
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Re: If i, a and b are integers, is 4(3b + 2) = 5a?  [#permalink]

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reto wrote:
If i, a and b are integers, is 4(3b + 2) = 5a?

(1) If i is divided by 5 the quotient is a and the remainder is 3
(2) If i is divided by 12 the quotient is b and the remainder is 11

My first instinct was to mark this question as C, as there are two unknown variables(a and b) and we can only get the values when we combine them.

But then GMAT is not about instinct, So to explain this, inline is the solution

Statement 1,
If i is divided by 5, quotient is a, remainder =3
This can be achieved when value of i = 8,13, 18

Statement 2,
If i is divided by 12, quotient is b, remainder =11
This can be achieved when value of i = 23,35,47

Now when you combine, you have to think of a value for i, when divided by 5 & 12, will give a remainder as 3 and 11 respectively.
i = 23 , solving will give a = 4 & b = 1, satisfying the question as Yes, 4 * 5 = 5* 4
i= 83, solving will give a = 16 & b = 6, satisfying the question as Yes, 4 * 20 = 5* 16

You can get an equation by combining both the statements as
N = 60ab + 23

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Re: If i, a and b are integers, is 4(3b + 2) = 5a?  [#permalink]

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reto wrote:
If i, a and b are integers, is 4(3b + 2) = 5a?

(1) If i is divided by 5 the quotient is a and the remainder is 3
(2) If i is divided by 12 the quotient is b and the remainder is 11

Target question: Is 4(3b + 2) = 5a?

Statement 1: If i is divided by 5 the quotient is a and the remainder is 3
ASIDE: There's a nice rule that says, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

So, we can take the given information and write: i = 5a + 3
Since we have no information about b, there's no way to answer the target question.
Statement 1 is NOT SUFFICIENT

Statement 2: If i is divided by 12 the quotient is b and the remainder is 11
We can write: i = 12b + 11
Since we have no information about a, there's no way to answer the target question.
Statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that i = 5a + 3
Statement 2 tells us that i = 12b + 11
Since both equations are set equal to i, we can write: 12b + 11 = 5a + 3
Subtract 3 from both sides to get: 12b + 8 = 5a
Factor left side to get: 4(3b + 2) = 5a
The answer to the target question is YES, it IS the case that 4(3b + 2) = 5a
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent

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_________________ Re: If i, a and b are integers, is 4(3b + 2) = 5a?   [#permalink] 23 Jan 2019, 07:00
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