ALGEBRA - REMAINDER THEOREM

Question

If positive integer x is divided by 5, the result is p and the remainder 3. If x is divided by 11, the remainder is 3 again, what is the remainder when p is divided by 11?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Here I'm having doubt when the Quotient of 1 number is getting divided by the Quotient of the other, then how to find remainder ? CAn you please clarify that ?

ArunSharma12 · Answer

x = 5p + 3
x = 11q + 3

taking LCM of the divisors from the above equations,
x = 55k + r, in order to find r you'll have to find the first number that satisfies the above two equations.

x = 5p + 3
x = 11q + 3: 3,14,25,36,47, 58 
58 when divided by 5 or 11 leaves remainder 3.

x = 55k + 3
but here we are not concerned with x, we want to find the value of p.
p = 11
rem(p/11)=0

CrackverbalGMAT · Answer

Hello Saurabh,

Clearly, there’s not much of a difficulty here. You just have to find the two quotients here, the moment you do that, you will be able to find the remainder. 
So what if a quotient is divided by another quotient? Instead of looking for a model of questions like this, don’t you think it’s better to stick to a fundamental approach of determining the dividend and the divisor so that you get the remainder??
  Bottom-line is – do not try to learn too many models, stick to the tried and tested fundamental approaches to make your life simpler.

However, there is one model that I want you to remember.

When you divide the same number by two different divisors and you obtain the same remainder in each case, the number can be expressed as 
Number = LCM (divisors)k + (Common remainder), where k is a non-negative integer.

In our case, the two divisors are 5 and 11; the common remainder is 3. Therefore, the number is,
N = LCM(5, 11)k + 3 OR N = 55k + 3 since the LCM of 5 and 11 is 55.
So, the possible values for N are 58, 113, 168 and so on.

If N = 58, p = 11. Remainder when p is divided by 11 is ZERO.

If N = 113, p = 22. Remainder when p is divided by 11 is ZERO.

In all case, we see that the value of p will always be a multiple of 11 and hence upon dividing by 11, the remainder obtained will always be ZERO.
 The correct answer option is A .

Hope that helps!

Bunuel · Answer

This question is discussed here: https://gmatclub.com/forum/if-positive- ... 30387.html Hope it helps.

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