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705-805 (Hard)|   Algebra|      
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Bunuel
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The total number of integer pairs (c, d) satisfying the equation c + d 25 Nov 2019, 01:37
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C
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Question Stats:

49% (01:36) correct 51% (01:30) wrong based on 279 sessions

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The total number of integer pairs (c, d) satisfying the equation c + d = cd is

A. 0
B. 1
C. 2
D. 3
E. 4


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C
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The total number of integer pairs (c, d) satisfying the equation c + d 25 Nov 2019, 04:02
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possible pairs
(0,0) & ( 2,2)
IMO C ; 2

Bunuel
The total number of integer pairs (c, d) satisfying the equation c + d = cd is

A. 0
B. 1
C. 2
D. 3
E. 4


Are You Up For the Challenge: 700 Level Questions
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The total number of integer pairs (c, d) satisfying the equation c + d 25 Nov 2019, 08:43
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Bunuel
The total number of integer pairs (c, d) satisfying the equation c + d = cd is

A. 0
B. 1
C. 2
D. 3
E. 4
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Couldn't think of any other pairs other than: 0,0 and 2,2 = 2

Ans (C)
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satya2029
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The total number of integer pairs (c, d) satisfying the equation c + d 26 Nov 2019, 00:29
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Bunuel
The total number of integer pairs (c, d) satisfying the equation c + d = cd is

A. 0
B. 1
C. 2
D. 3
E. 4


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c+d=cd
c/cd+d/cd=1
1/c+1/d=1
(c,d)=(2,2)
B:)
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The total number of integer pairs (c, d) satisfying the equation c + d 26 Nov 2019, 03:28
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c+d=cd
d=cd-c
d=c(d-1)
c=d/(d-1)
Now when d=0 c=0
d=1 c=not defined
And when d=2 c=2
d=3 c=3/2=1.5
Thus, For every other integer value of d apart from 0 & 2; c is never an integer.
Therefore only two integer pairs of (c,d) are possible i.e.(0,0) and (2,2).

Answer: C

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The total number of integer pairs (c, d) satisfying the equation c + d 10 Jun 2022, 16:33
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If we move everything to one side and set the equation equal to 0, we can look for a factorable opportunity.

CD — C — D = 0

C * (D - 1) — D = 0

—adding +1 to both sides of the equation allows us to put the terms within two expressions that multiply to a constant

C * (D - 1) — D + 1 = +1

C * (D - 1) - 1 (D - 1) = 1

(D - 1) (C - 1) = 1

Now we have two factors, the product of which = positive 1

There are two ways to make this true:

(+1) * (+1) = 1
Or
(-1) * (-1) = 1

so we can set the factors equal to the numbers in each possibility

Possibility #1:
C = 0 , D = 0

Possibility #2:
C = 2 , D = 2

Answer C
2

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The total number of integer pairs (c, d) satisfying the equation c + d 26 Jun 2024, 08:26
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