If $$\sqrt{x}\ $$ and $$\sqrt{y}$$ are nonzero

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If $$\sqrt{x}\ $$ and $$\sqrt{y}$$ are nonzero integers, what is the remainder when x is divided by y?
(1) $$\sqrt{x}\ -\ \sqrt{y}$$ = 1
(2) x-y is a positive integer.

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by fskilnik@GMATH » Fri Jan 18, 2019 5:27 pm
DivyaD wrote:If sqrt{x} and sqrt{y} are nonzero integers, what is the remainder when x is divided by y?
(1) $$\sqrt{x}\ -\ \sqrt{y} = 1$$
(2) x-y is a positive integer.


$$\left\{ \matrix{
\,x = {M^2}\,,\,\,M\mathop \ge \limits^{\left( * \right)} 1\,\,{\mathop{\rm int}} \hfill \cr
\,y = {N^2}\,,\,\,N\mathop \ge \limits^{\left( * \right)} 1\,\,{\rm{int}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\left( * \right)\,\,{\rm{WLOG}}\,\,\,\left( {{\rm{without}}\,\,{\rm{loss}}\,\,{\rm{of}}\,\,{\rm{generality}}} \right)$$

$$?\,\,\,\,:\,\,\,\,x\,\,{\rm{over}}\,\,y\,\,\,\,{\rm{remainder}}$$

$$\left( 1 \right)\,\,1 = \sqrt x - \sqrt y = M - N$$
$$\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {M,N} \right) = \left( {2,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {x,y} \right) = \left( {4,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 0 \hfill \cr
\,{\rm{Take}}\,\,\left( {M,N} \right) = \left( {3,2} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {x,y} \right) = \left( {9,4} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 1 \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{INSUFF}}.$$

$$\left( 2 \right)\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {x,y} \right) = \left( {4,1} \right)\,\,\,\,\,\, \hfill \cr
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {x,y} \right) = \left( {9,4} \right)\,\,\,\,\,\, \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {\rm{E}} \right)$$


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

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Fabio.
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by Jay@ManhattanReview » Sun Jan 20, 2019 8:47 pm
DivyaD wrote:If √x and √y are nonzero integers, what is the remainder when x is divided by y?

(1) √x - √y = 1
(2) (x - y) is a positive integer.
Given: √x and √y are nonzero integers

We have to find out the remainder when x is divided by y.

Let's take each statement one by one.

(1) √x - √y = 1

Case 1: Say x = 4 and y = 1, then we have √4 - √1 = 1. The remainder when x = 4 is divided by y = 1 is 0.
Case 2: Say x = 9 and y = 4, then we have √9 - √4 = 1. The remainder when x = 9 is divided by y = 4 is 1.

No unique answer. Insufficient.

(2) (x - y) is a positive integer.

Certainly insufficient.

(1) and (2) together

Since in both the cases discussed in Statement 1, we have (x - y), a positive integer. Thus, both the statements together are insufficient.

The correct answer: E

Hope this helps!

-Jay
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