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If each of d, q, and r is a positive integer such that \(dq + r = 3\), there are 3 possible cases.

1) r=1; d=2; q=1
2) r=1; d=1; q=2
3) r=2; d=1; q=1

Statement 1-

\(\frac{r}{d} <\frac{2}{3}\)

case 1- \(\frac{r}{d} = \frac{1}{2} <\frac{2}{3}\)

case 2- \(\frac{r}{d} = 1 > \frac{2}{3}\) (Rejected)

case 3- \(\frac{r}{d} = 2 > \frac{2}{3}\) (Rejected)

d=2
Sufficient

Statement 2-

\(\frac{q}{r}<2\)

Case 1- \(\frac{q}{r}=1\)<2; d=2

Case 2- \(\frac{q}{r}=2\) (Rejected)

Case 3- \(\frac{q}{r} =\frac{1}{2}\) {less than 2 but not an integer} (Rejected)

sufficient



Bunuel
If each of d, q, and r is a positive integer such that \(dq + r = 3\), what is the value of d?


(1) The number \(\frac{r}{d}\) is less than 2/3.

(2) The integer \(\frac{q}{r}\) is less than 2.
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dq+r=3
2*1+1 = 3
1*1+2 = 3

(1) r/d < 2/3
so r is a smaller integer, and d is a bigger integer. That means r must be 1, and d must be 2. SUFFICIENT

(2) q/r <2
so q is larger than r, but is less than two. That means r can't be 2. So q must be 1, and r must be 1, therefore d must be 2. SUFFICIENT
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Bunuel
If each of d, q, and r is a positive integer such that \(dq + r = 3\), what is the value of d?


(1) The number \(\frac{r}{d}\) is less than 2/3.

(2) The integer \(\frac{q}{r}\) is less than 2.

Critical info


a) each of d, q, and r is a positive integer
b) \(dq + r = 3\)

Inference


Each of d, q and r have to be less than 3, so possible values are 1,2
Also one of them will be 2 and others will be 1 each.

Statements



(1) The number \(\frac{r}{d}\) is less than 2/3.
Now, r and d have to be only 1 and 2, so possible values of r/d is 2/1 or 1/2.
\(\frac{1}{2}<\frac{2}{3}...\frac{r}{d}=\frac{1}{2}.........d=2\)
Sufficient

(2) The integer \(\frac{q}{r}\) is less than 2
Now, r and q have to be only 1 and 2, so possible values of q/r is 2/1 or 1/2 or 1/1.
2/1=2, so eliminate as q/r<2
1/2 is NOT an integer
Only possibility is 1/1, so q=r=1. Hence, d=2.
Sufficient

D
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