Kinshook wrote:
If x & y are integers, how many ordered pairs (x,y) satisfy the equation (|x-2|-12)(|y+4|-36)=72?
A. 48
B. 24
C. 72
D. 144
E. 36
Given: x & y are integers
Asked: How many ordered pairs (x,y) satisfy the equation (|x-2|-12)(|y+4|-36)=72?
\(72 = 2^3*3^2\)
72 has 4*3 = 12 factors
Factors of 72 = {1,2,3,4,6,8,9,12,18,24,36,72}
If x & y are positive integers, we will have 12 cases
72=1*7272= (|x-2|-12)(|y+4|-36)
|x-2|-12=1 => |x-2|=12+1=13 =>x=2+-13 => x=15 or x=-11
|y+4|-36=72 =>|y+4|=36+72=108 =>y=-4+-108 => y=112 or y=-104
This case will give 4 ordered pairs = (15,112),(-11,112),(15,-104) & (-11,-104)
72=2*3672= (|x-2|-12)(|y+4|-36)
|x-2|-12=2 => |x-2|=12+2=14 =>x=2+-14 => x=16 or x=-12
|y+4|-36=36 =>|y+4|=36+36=72 =>y=-4+-72 => y=-76 or y=68
This case will give 4 ordered pairs = (16,68),(-12,-76),(16,68) & (-12,-76)
72=3*2472= (|x-2|-12)(|y+4|-36)
|x-2|-12=3 =>|x-2|=12+3=15 => x=2+-15 => x=17 or x=-13
|y+4|-36=24 =>|y+4|=36+24=60 =>y=-4+-60 => y=-64 or y=56
This case will give 4 ordered pairs = (17,56),(-13,-64),(17,56) & (-13,-64)
72=4*1872= (|x-2|-12)(|y+4|-36)
|x-2|-12=4 => |x-2|=12+4=16 => x=2+-16 => x=18 or x=-14
|y+4|-36=18 =>|y+4|=36+18= 54 => y=-4+-54 => y=-58 or y=50
This case will give 4 ordered pairs = (18,50),(-14,-58),(18,50) & (-14,-58)
72=6*1272= (|x-2|-12)(|y+4|-36)
|x-2|-12=6 => |x-2|=12+6=18 => x=2+-18 => x=20 or x=-16
|y+4|-36=12 =>|y+4|=36+12= 48 => y=-4+-48 => y=-52 or y=44
This case will give 4 ordered pairs = (20,44),(-16,-52),(20,44) & (-16,-52)
72=8*972= (|x-2|-12)(|y+4|-36)
|x-2|-12=8 => |x-2|=12+8=20 => x=2+-20 => x=22 or x=-18
|y+4|-36=9 =>|y+4|=36+9= 45 => y=-4+-45 => y=-49 or y=41
This case will give 4 ordered pairs = (22,41),(-18,-49),(22,41) & (-18,-49)
72=9*872= (|x-2|-12)(|y+4|-36)
|x-2|-12=9 => |x-2|=12+9=21 => x=2+-21 => x=23 or x=-19
|y+4|-36=8 =>|y+4|=36+8= 44 => y=-4+-44 => y=-48 or y=40
This case will give 4 ordered pairs = (23,40),(-19,-48),(23,40) & (-19,-48)
72=12*672= (|x-2|-12)(|y+4|-36)
|x-2|-12=12 => |x-2|=12+12=24 => x=2+-24 => x=26 or x=-22
|y+4|-36=6 =>|y+4|=36+6= 42 => y=-4+-42 => y=-46 or y=38
This case will give 4 ordered pairs = (26,38),(-22,-46),(26,38) & (-22,-46)
72=18*472= (|x-2|-12)(|y+4|-36)
|x-2|-12=18 => |x-2|=12+18=30 => x=2+-30 => x=32 or x=-28
|y+4|-36=4 =>|y+4|=36+4= 40 => y=-4+-40 => y=-44 or y=36
This case will give 4 ordered pairs = (32,36),(-28,-44),(32,36) & (-28,-44)
72=24*372= (|x-2|-12)(|y+4|-36)
|x-2|-12=24 => |x-2|=12+24=36 => x=2+-36 => x=38 or x=-34
|y+4|-36=3 =>|y+4|=36+3= 39 => y=-4+-39 => y=-43 or y=35
This case will give 4 ordered pairs = (38,35),(-34,-43),(38,35) & (-34,-43)
72=36*272= (|x-2|-12)(|y+4|-36)
|x-2|-12=36 => |x-2|=12+36=48 => x=2+-48 => x=50 or x=-46
|y+4|-36=2 =>|y+4|=36+2= 38 => y=-4+-38 => y=-42 or y=34
This case will give 4 ordered pairs = (50,34),(-46,-46),(50,34) & (-46,-46)
72=72*172= (|x-2|-12)(|y+4|-36)
|x-2|-12=72 => |x-2|=12+72=84 => x=2+-84 => x=86 or x=-82
|y+4|-36=1 =>|y+4|=36+1= 37 => y=-4+-37 => y=-41 or y=33
This case will give 4 ordered pairs = (86,33),(-82,-41),(86,33) & (-82,-41)
These 12 cases when x & y are positive integers will give 12*4 = 48 ordered pairs
Now let us consider negative integer cases:-
72=(-1)*(-72)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-1 => x=2+-11 => x=13 or x=-9
|y+4|-36=-72 => |y+4| = 36-72 = -36 => NOT FEASIBLE
This case will give 0 ordered pairs
72=(-2)*(-36)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-2 => x=2+-10 => x=12 or x=-8
|y+4|-36=-36 => |y+4| = 0 => y = -4
This case will give 2 ordered pairs (12,-4) & (-8,-4)
This case will give 2 ordered pairs
72=(-3)*(-24)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-3 => x=2+-9 => x=11 or x=-7
|y+4|-36=-24 => |y+4| = 36-24 = 12 => y = -4+-12 => y=8 or y=-16
This case will give 4 ordered pairs (11,8),(11,-16),(-7,8) & (-7,-16)
This case will give 4 ordered pairs
72=(-4)*(-18)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-4 => x=2+-8 => x=10 or x=-6
|y+4|-36=-18=> |y+4| = 36-18 = 18 => y = -4+-18 => y=14 or y=-22
This case will give 4 ordered pairs
72=(-6)*(-12)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-6 => x=2+-6 => x=8 or x=-4
|y+4|-36=-12=> |y+4| = 36-12 = 24 => y = -4+-24 => y=20 or y=-28
This case will give 4 ordered pairs
72=(-8)*(-9)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-8 => x=2+-4 => x=6 or x=-2
|y+4|-36=-9=> |y+4| = 36-9 = 27 => y = -4+-27 => y=23 or y=-31
This case will give 4 ordered pairs
72=(-9)*(-8)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-9 => |x-2|= 12-9=3 => x=2+-3 => x=5 or x=-1
|y+4|-36=-8=> |y+4| = 36-8 = 28 => y = -4+-28 => y=24 or y=-32
This case will give 4 ordered pairs
72=(-12)*(-6)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-12 => |x-2|= 12-12=0 => x= 2
|y+4|-36=-6=> |y+4| = 36-6 = 30 => y = -4+-30 => y=26 or y=-34
This case will give 2 ordered pairs
72=(-18)*(-4)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-18 => |x-2|= 12-18=-6 => NOT FEASIBLE
|y+4|-36=-4=> |y+4| = 36-4 = 32 => y = -4+-32 => y=28 or y=-36
This case will give 0 ordered pairs
72=(-24)*(-3)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-24 => |x-2|= 12-24=-12 => NOT FEASIBLE
|y+4|-36=-3=> |y+4| = 36-3 = 33 => y = -4+-33 => y=29 or y=-37
This case will give 0 ordered pairs
72=(-36)*(-2)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-36 => |x-2|= 12-36=-24 => NOT FEASIBLE
|y+4|-36=-2=> |y+4| = 36-2 = 34 => y = -4+-34 => y=30 or y=-38
This case will give 0 ordered pairs
72=(-72)*(-1)72= (|x-2|-12)(|y+4|-36)
|x-2|-12=-72 => |x-2|= 12-72=-60 => NOT FEASIBLE
|y+4|-36=-1=> |y+4| = 36-1 = 35 => y = -4+-35 => y=31 or y=-39
This case will give 0 ordered pairs
Total ordered pairs for negative integers = 2+ 5*4 +2 = 24
Total integer ordered pairs = 48 + 24 = 72
IMO A