Quantitative Comparison - Number theory

Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.

N = 1323*1325*1327*1329

Quantity A : Rem[N/13] + 1

Quantity B : Rem[N/11]

where Rem[A/B] is defined as the remainder obtained when A is divided by B.

Solution: (C)

Explanation:

The solution to this question becomes very simple with the understanding of a simple concept involving remainders.

If N = x₁ * x₂ * x₃ * ……… * x_n

And if

Rem[x₁/k] = r₁

Rem[x₂/k] = r₂ and so on…..

Then,

Rem[N/k] = Rem[(r₁ * r₂ * r₃ * ………..* r_n)/k]

This can be continued recursively until we are left with a single remainder.

Now, in this question,

N = 1323*1325*1327*1329

Quantity A: k = 13

Therefore,

Rem[1323/13] = 10

Rem[1325/13] = 12

Rem[1327/13] = 1

Rem[1329/13] = 3

Hence Rem[N/13] = Rem[(10*12*1*3)/13]

                              = Rem[360/13]

                              = 9

Similarly,

Quantity B: k = 11

Therefore,

Rem[1323/11] = 3

Rem[1325/11] = 5

Rem[1327/11] = 7

Rem[1329/11] = 9

Hence Rem[N/11] = Rem[(3*5*7*9)/11]

                             = Rem[945/11]

                             = 10

Or, to illustrate how this can be used recursively,

      Rem[N/11] = Rem[(3*5*7*9)/11]

                         = Rem{[Rem(27/11)*Rem(35/11)]/11}

                         = Rem{[5*2]/11}

                         = Rem[10/11]

                         = 10