If a is a positive integer and 81 divided by a

[swerve]

If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

1) The remainder when a is divided by 40 is 0.
2) The remainder when 40 is divided by a is 40.

The OA is B

Source: Manhattan Prep

[Brent@GMATPrepNow]

If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

1) The remainder when a is divided by 40 is 0.
2) The remainder when 40 is divided by a is 40.

Target question: What is the value of a?

Given: 81 divided by a results in a remainder of 1
In other words, 81 is 1 greater than some multiple of a
This means that 80 is some multiple of a
Another way to say this is: a is a divisor of 80
The divisors of 80 are: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
So, the possible values of a are: 2, 4, 5, 8, 10, 16, 20, 40, 80
ASIDE: I omitted 1 from the list, since it does not satisfy the condition of getting a remainder of 1 when 81 is divided by 1.

Statement 1: The remainder when a is divided by 40 is 0.
The possible values of a are: 2, 4, 5, 8, 10, 16, 20, 40, 80
As you can see, there are two possible values of a satisfy statement 1.
Case a: a = 40. This works, because 40 divided by 40 leaves remainder 0. In this case, the answer to the target question is a = 40
Case b:a = 80. This works, because 80 divided by 40 leaves remainder 0. In this case, the answer to the target question is a = 80
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The remainder when 40 is divided by a is 40.
The possible values of a are: 2, 4, 5, 8, 10, 16, 20, 40, 80
As you can see, ONLY ONE possible value of a satisfies statement 2.
When a = 80, we get a remainder of 40 when 40 is divided by 80
So, the answer to the target question is a = 80
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

[Jay@ManhattanReview]

If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

1) The remainder when a is divided by 40 is 0.
2) The remainder when 40 is divided by a is 40.

The OA is B

Source: Manhattan Prep

Given: a is a positive integer and 81 divided by a results in a remainder of 1.

We have to find out the value of a.

From the given information, we know that a is a factor of 80.

Let's take each statement one by one.

1) The remainder when a is divided by 40 is 0.

=> a is a factor of 40; however, we can't know the unique value of a. Insufficient.

2) The remainder when 40 is divided by a is 40.

=> a > 40.

Since only one factor of 80 is greater than 40, i.e. 80, we have a = 80. Sufficient.

The correct answer: B

Hope this helps!

-Jay
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[fskilnik@GMATH]

If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

1) The remainder when a is divided by 40 is 0.
2) The remainder when 40 is divided by a is 40.
Source: Manhattan Prep

Very nice conceptual problem!

$$a \ge 1\,\,{\mathop{\rm int}} \,\,\,\,\,\left( * \right)$$
$$81 = M \cdot a + 1\,\,,\,\,M\mathop \ge \limits^{\left( * \right)} 1\,\,{\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,M \cdot a = 80\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,a \le 80\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{positive}}\,\,{\rm{divisor}}\,\,{\rm{of}}\,\,80\,\,\,\,\left( {**} \right)$$
$$? = a$$

$$\left( 1 \right)\,\,a = 40J\,\,,\,\,\,J\mathop \ge \limits^{\left( * \right)} 1\,\,\,{\mathop{\rm int}} \left\{ \matrix{
\,{\rm{Take}}\,\,J = 1\,\,\,\,\left( {M = 2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 40\,\,\,\,{\rm{viable}} \hfill \cr
\,{\rm{Take}}\,\,J = 2\,\,\,\,\left( {M = 1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 80\,\,\,\,{\rm{viable}} \hfill \cr} \right.$$

$$\left( 2 \right)\,\,40 = N \cdot a + \underline {40} \,\,,\,\,N\,\,{\mathop{\rm int}} \,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\left\{ \matrix{
\,a\,\, > \,\,\underline {40} \hfill \cr
\,N = 0 \hfill \cr} \right.\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,a = 80\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.$$


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.