M21-35

Question

If  x  is a positive integer, what is the remainder when  x  is divided by 3?

(1) The remainder when  x  is divided by 4 is 1
 
(2) The remainder when  x  is divided by 7 is 6

Bunuel · Accepted Answer

Official Solution:

If  x  is a positive integer, what is the remainder when  x  is divided by 3? 
 
(1) The remainder when  x  is divided by 4 is 1.
 
Given:  x=4q+1 , so  x  could be: 1, 5, 9,  13 , 17, 21, 25, 29, 33, 37,  41 , ... This information is not sufficient to determine the remainder when  x  is divided by 3.
 
(2) The remainder when  x  is divided by 7 is 6.
 
Given:  x=7p+6 , so  x  could be: 6,  13 , 20, 27, 34,  41 , ... This information is not sufficient to determine the remainder when  x  is divided by 3.
 
(1)+(2) If  x=13  then the remainder is 1 when divided by 3, but if  x=41  then the remainder is 2 when divided by 3.Not sufficient.

Answer: E

samuraijack256 · Answer

Thanks for the explanation Bunuel. I too used the same method.

However, I was wondering if there is any faster method to determine the two numbers (13 and 41).

In the exam, my intent is to save as much time as possible and move on to the next question. In that state of mind, I will be inclined to just test the (say) first 5 numbers of the sequence. In the case of (1) above, that will be just 1, 5, 9, 13, 17
and for (2), this will be 6,13, 20, 27, 34.

Clearly, I will have missed the important numbers (13 and 41) which are key to swinging towards the correct answer.
Any suggestions?

chetan2u · Answer

Hi,
 The faster method in these Qs is to find the first common multiple and, thereafter, add the LCM of two numbers..

In this case when we have got the first number as 13.. the next number will be 13 + lcm of 4 and 7=13+28=41..
the next 41+28=69 and so on...

so 13 gives remainder 1
 41 gives remainder 2
 69 gives remainder 0... and so on
hope it helped..

chetan2u · Answer

another way would be to equate two eq from two statements..

x=4q+1 ,  x=7p+6 ...
 4q+1=7p+6 ....
 4q=7p+5 ....

now substitute p as 1,3,5... as p has to be odd to make the eq even ..
where we getan integer value of q... those are the numbers we rae looking for..
for ex
p=1,q=3... so number=4q+1=13..
p=3, q is not an int..
p=5, q=10 number =4q+1=41.. and so on

anka1994 · Answer

I think this is a  high-quality  question and I agree with explanation.

efr · Answer

Hi Bunuel- Can you please confirm if my logic is right while doing this exercise?

n = 4k + 1 -> Insuf

n = 7k + 6 -> Insuf

n = 28k + 13 (13 is the frist common number of 4k+1 and 7k+6).

As 28k is not divide by 3, the Reminder will change.

Thanks in advance!

Bunuel · Answer

Correct with small but crucial correction: quotient in all three cases (highlighted) will not be the same so you should not sue one variable for them (use p, q, k instead).

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

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M21-35

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