+0  
 
0
8
3
avatar+1359 

Find the number of ordered pairs $(a,b)$ of integers such that
\frac{a + 2}{a + 1} = \frac{b}{8}.

 Apr 2, 2024

Best Answer 

 #2
avatar+129690 
+1

(a + 2) / (a + 1)  =  b /  8

 

b =   8 (a + 2)  / (a + 1)

 

The function has a  horizontal asymptote at  b = 8

 

a          b

0         16

1         12

3         10

7          9

-2        0

-3        4

-5        6

-9        7 

 

cool cool cool                                               

 Apr 3, 2024
 #1
avatar+1696 
0

Multiplying both sides by 8(a+1) (which is nonzero since a is an integer), we get: $8(a+2)=b(a+1).$This forces b to be a multiple of 8. We can rewrite the equation as: $b=8k⋅a+1a+2​=8k(1+a+11​)$Since b is an integer, k must also be an integer.

If a=−1, then the fraction becomes undefined, so we must reject this possibility. Otherwise, a+11​ is an integer, so a+1 must divide 1. The only two possibilities are a=0 and a=−2.

If a=0, then b=16k, which means that k can be any integer. So, in this case, there are infinitely many solutions (a,b).

If a=−2, then b=−8k, which means that k must be nonpositive. The only nonpositive integer that divides 8 is −8, so k=−8 and b=64. Thus, the only solution in this case is (−2,64).

In conclusion, there are ∞​ solutions if a=0 and exactly 1​ solution if a=−2.

 Apr 2, 2024
 #2
avatar+129690 
+1
Best Answer

(a + 2) / (a + 1)  =  b /  8

 

b =   8 (a + 2)  / (a + 1)

 

The function has a  horizontal asymptote at  b = 8

 

a          b

0         16

1         12

3         10

7          9

-2        0

-3        4

-5        6

-9        7 

 

cool cool cool                                               

CPhill Apr 3, 2024
 #3
avatar+129690 
+1

Another way

8(a + 2) = b (a + 1)

 

8a + 16 - ab - b  =  0

 

8a + 16 - ab - b + (-1 * 8)  =  (-1*8)

 

8a + 8  - ab -b  =  -8

 

8 ( a + 1)  - b ( a + 1)  =  -8

 

(a + 1) ( 8 - b)   =  -8

 

Factors  of -8             a        b

-2  4                          -3       4

4  -2                           3      10

-4  2                          -5      6

2  -4                           1     12

1  -8                           0     16

-8  1                          -9      7

-1  8                          -2      0

8  -1                          7       9

 

cool cool cool

 Apr 3, 2024

5 Online Users

avatar
avatar