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If each of d, q, and r is a positive integer such that dq + r = 3, wha

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Bunuel
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If each of d, q, and r is a positive integer such that dq + r = 3, wha [#permalink] New post   20 Mar 2020, 04:58
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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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Re: If each of d, q, and r is a positive integer such that dq + r = 3, wha [#permalink] New post   20 Mar 2020, 05:06
Bunuel wrote:
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.


Asked: If each of d, q, and r is a positive integer such that \(dq + r = 3\), what is the value of d?
2 out of (d, q and r) = 1 and 1 out of (d, q and r) = 2


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

(2) The integer \(\frac{q}{r}\) is less than 2.
q/r < 2
q/r = 1
q = r = 1
d = 2
SUFFICIENT

IMO D
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Re: If each of d, q, and r is a positive integer such that dq + r = 3, wha [#permalink] New post   05 Apr 2020, 14:26
Bunuel wrote:
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.


d,q, r all can be either 1 or 2.
1) When d =2, q and r have to be 1 to hold the equation given in the stem true. When d =1, r can be either 2 or 1 depending on the value of q. Different answers possible. Insuff
2) q =1, r =2. That means d =1. Suff.
B is the answer
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Re: If each of d, q, and r is a positive integer such that dq + r = 3, wha [#permalink] New post   05 Apr 2020, 16:09
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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 wrote:
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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Re: If each of d, q, and r is a positive integer such that dq + r = 3, wha [#permalink] New post   05 Apr 2020, 23:11
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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Re: If each of d, q, and r is a positive integer such that dq + r = 3, wha [#permalink] New post   05 Apr 2020, 23:37
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Bunuel wrote:
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

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Re: If each of d, q, and r is a positive integer such that dq + r = 3, wha [#permalink]
05 Apr 2020, 23:37
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