If a is a positive integer and 81 divided by a results in a

Question

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

Bunuel · Accepted Answer

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

Given:  81=qa+1  -->  qa=80  --> so,  a  is a factor of 80: 2, 4, 5, 8, 10, 16, 20, 40, or 80.

(1) The remainder when a is divided by 40 is 0 --> this basically tells us that  a  is a multiple of 40, so  a  could be either 40 or 80 (from the above). Not sufficient.

(2) The remainder when 40 is divided by a is 40 --> this basically means that  a  is greater than 40 (for example, the remainder upon division 40 by 50 is 40). Now, the only factot of 80 which is greater than 40 is 80 itself. So,  a=80 . Sufficient.

Answer: B.

Hope it's clear.

kashishh · Answer

Dear Bunuel,
 i am not able to understand the st. 2 analysis. (ofcourse i understood what you have written.)
 one doubt remains like: if it is said x/y leaves a remainder of 10, can i consider values of x/y as 10/20 and like that?

Bunuel · Answer

Yes, 10 divided by 20 yields the remainder of 10.

THEORY:
 Positive integer  a  divided by positive integer  d  yields a reminder of  r  can always be expressed as  a=qd+r , where  q  is called a quotient and  r  is called a remainder, note here that  0\leq{r}<d  (remainder is non-negative integer and always less than divisor).

For example positive integer n is divided by 25 yields the remainder of 13 can be expressed as:  n=25q+13 . Now,  the lowest value of  q  can be zero  and in this case  n=13  --> 13 divided by 25 yields the remainder of 13.  Generally when divisor (25 in our case) is more than dividend (13 in our case) then the reminder equals to the dividend . For example:
3 divided by 24 yields a reminder of 3 -->  3=0*24+3 ;
or:
5 divided by 6 yields a reminder of 5 -->  5=0*6+5 ,

Hope it's clear.

ichauhan.gaurav · Answer

Thanks for the solution I have one doubt ... as per the 2nd statement when 40 is divided by a remainder is 40 i.e. 40 = aq +40 or aq =0 ? can you please explain this also ..

Bunuel · Answer

Yes, "the remainder when 40 is divided by a is 40" can be written as  40=ap+40  -->  ap=0 . What is the point of confusion here?

GyMrAT · Answer

Given a > 0 & 81 = ak + 1

a = ?

Statement 1:

The remainder when a is divided by 40 is 0

a = 40p = 40 or 80
to satisfy 81 = ak + 1

Statement 1 is Not Sufficient.

Statement 2:

The remainder when 40 is divided by a is 40

40 = aq + 40, hence a > 40 & has to satisfy 81 = ak + 1

Hence a = 80

Statement 2 is Sufficient.

Answer B.

Thanks,
GyM

BrentGMATPrepNow · Answer

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

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fskilnik · Answer

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.

OmerKor · Answer

Hi Bunuel,

Following up this Question b/c I tried solving it today.

Why does “a” can’t be multiple of 81 such as 162,243,324..
81/81x2 = 1/2. R=1 
81/81x3 = 1/3. R=1

Posted from my mobile device

Bunuel · Answer

You cannot reduce like that when calculating a remainder. For example, 12 divided by 8 yields a remainder of 4. However, if you reduce each number by 4, you get 3 divided by 2, which results in a remainder of 1, not 4. Thus, the remainder was reduced by the same factor.

Hence, 81 divided by 162 results in a remainder of 81. Generally, when the divisor is greater than the dividend, the remainder is equal to the dividend.

6. Remainders

Theory   
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 Remainders: Tips and hints 
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