(|x| + 3) (|y| - 4) = 12. How many pairs of integers (x,y) satisfy

@Dilesh4096... Same question from me

Posted from my mobile device

Hi Dillesh4096,

(|x|+3) ≥3 and thus (|y|-4) has to be positive in any combination in order to satisfy (|x|+3)(|y|-4)= 12,

(|x|+3) can take four values: 3, 4, 6 and 12. Correspondingly, (|y|-4) must pair with following values: 4, 3, 2, 1.

(|x|+3)(|y|-4)
=3*4... x=0,y=+/-8, number of solution=1*2=2
=4*3...x=+/-1,y=+/-7, number of solution=2*2=4
=6*2...x=+/-3,y=+/-6, number of solution=2*2=4
=12*1...x=+/-9,y=+/-5, number of solution=2*2=4

Thus, total number of pairs of integers (x,y) that satisfy the equation = 2+4+4+4=14
Answer is (D)

Hit that +1 kudo if you like my solution

Posted from my mobile device

I used the following concepts to solve this problem
1.
If |x|=0, then there is only 1 value possible for x, that is x=0.
If |x|=k, where k is non-zero positive real number, then there are 2 values of x are possible, that are +k and -k.

2.
If |x|=m, and |y|=n. Hence both x and y can have 2 values. total number of combinations possible for (x, y)=2*2=4
(x, y)= (m, n), (-m,n), (m, -n) and (-m, -n)

You don't have to to find the solutions in all cases, as |x| and |y| are taking distinct values in all 4 cases. So, there can't be any overlapping possible.
Just use the combinatorics, helping you to save some time.

[/quote]


Can you Please explain this bit :
number of solutions=1*2=2
number of solutions=2*2=4
number of solutions=2*2=4
number of solutions=2*2=4[/quote]

Detailed Solution:

(|x| + 3) (|y| - 4) = 12
Possible values of 12 can be 12*1, 6*2, 4*3

Note that, minimum value of (|x| + 3) is 3 as lxl CANNOT be less than 0
--> Possible values of (|x| + 3) are 3, 4, 6, 12

Case 1: When lxl + 3 = 3 ; (|y| - 4) = 4
--> lxl = 0
--> (lyl - 4) = 4
--> lyl = 8
--> y = ±8
Possible values of (x, y) = (0,8) & (0, -8) -- 2 solutions

Case 2: When lxl + 3 = 4 ; (|y| - 4) = 3
--> lxl = 1
--> x = ±1
--> (lyl - 4) = 3
--> lyl = 7
--> y = ±7
Possible values of (x, y) = (1,7), (1, -7), (-1, 7) & (-1, -7) -- 4 solutions

Case 3: When lxl + 3 = 6 ; (|y| - 4) = 2
--> lxl = 3
--> x = ±3
--> (lyl - 4) = 2
--> lyl = 6
--> y = ±6
Possible values of (x, y) = (3,6), (3, -6), (-3, 6) & (-3, -6) -- 4 solutions

Case 4: When lxl + 3 = 12 ; (|y| - 4) = 1
--> lxl = 9
--> x = ±9
--> (lyl - 4) = 1
--> lyl = 5
--> y = ±5
Possible values of (x, y) = (9,5), (9, -5), (-9, 5) & (-9, -5) -- 4 solutions

Total number of solutions = 2 + 4 + 4 + 4 = 14

Hope it's clear!

Asked: (|x| + 3) (|y| - 4) = 12. How many pairs of integers (x,y) satisfy this equation ?

12 = 2^2*3 = 1*12 = 2*6 = 3*4

|x| + 3 = 1 ; |y|-4 = 12; 0 solutions
|x| + 3 = 12 ; |y|-4 = 1; |x| = 9; |y| = 5; 4 solutions
|x| + 3 = 2 ; |y|-4 = 6; 0 solutions
|x| + 3 = 6 ; |y|-4 =2; |x| = 3; |y| = 6; 4 solutions
|x| + 3 = 3 ; |y|-4 = 4; |x| = 0; |y| = 8; 2 solutions
|x| + 3 = 4 ; |y|-4 =3; |x| = 1; |y| = 7; 4 solutions

Total solutions = 4*3 + 2 = 14 solutions

IMO D