If x & y are integers, how many solution pairs (x,y) satisfy the equat

Question

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

Archit3110 · Accepted Answer

giving a try ; 
total factors of 72 ; 2^3*3^2 ; 4*3 ; 12
so answer has to be multiple of 12
from given expression (|x-2|-12)(|y+4|-36)=72
for values withing modulus we can get 4 pairs i.e x,y ( +,+) ( -,-) (+,-) (-,+)
also value either of x or y is 0 so ( 0,26) & ( 12,0)
so total possible pairs 6 * factor ; 6*12 ; 72
IMO C

kshitijhbti · Answer

72 = {1,2,3,4,6,8,9,12,18,24,72}

A x B = 72

Case 1

A=1 B=72
Then (x,y) will be (13,104),(13,-112),(-11,104) and (-11,-112)

and so on for A=72 and B=1

Total such 12 A B pair cases will be there

And 4 integral pair will come as a result of each A B pair

Hence 48 is the answer in my opinion.

Posted from my mobile device

Kinshook · Answer

A very good attempt but x & y may be negative integers. 
Therefore, consider that case.

Kinshook · Answer

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*72 
72= (|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*36 
72= (|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*24 
72= (|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*18 
72= (|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*12 
72= (|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*9 
72= (|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*8 
72= (|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*6 
72= (|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*4 
72= (|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*3 
72= (|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*2 
72= (|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*1 
72= (|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

devangpandey007 · Answer

Is there a shorter way to attempt this question?

Kinshook · Answer

You need to check values of modulus is positive for each case and need not solve final values of x and y.

Posted from my mobile device

Preetiv · Answer

First determine the number of factors of 72: 2^3*3^2 => (3+1)*(2+1)=12 factors
Then, use the chart below to determine the zones for positivity of negativity of combined modulus on the number line:
______region 1____ region 2_____region 3
______________-4________2_____________
x-2 ---------------------------0++++++++++
y+4 -------------0++++++++++++++++++

Since there are 3 regions, (x,y) couple can take 3 different set of values for a couple of positive factors of 72. So we have 3*12 possibilities of couples. 
Finally , the factors of 72 can be both positive or both negative, so multiply the number of couples found for positive factors by 2 =>3*12*2=72 couple possibilities.

Can you help me to see if this reasoning is correct?

Aarya_San · Answer

ok so here was my attempt, maybe I'm missing something:

72 -> {(8,9),(3,24),(6,12),(36,2),(18,4),(72,1)}

There are 6 pairs. Now these pairs can be interchanged eg: 8*9 or 9*8, therefore multiple 6 with 2 = 12 ways

Now for each case, x can be positive or negative, eg: let's say for (-8*-9) -> x = 6 or -2 -> (|6-2| - 12) = -8 or (|-2-2| -12) = -8. (Note: this idea will work for each pair that we have). So multiple 12 by 2 = 24 ways

Now the exact same scenario will play out for y as well: Multiple 24*2 = 48.

Most would think the question is done by now, but unfortunately we still have the negative numbers left.

Like the example I took above, 8*9 and 9*8 can give us 72 but -8*-9 or -9*-8 can also give us 72. So let's multiple 48 by 2 = 96  (This is a mistake!)

This problem with this approach is that we over look the fact that the equation is set up in such a way that some negative numbers aren't possible. So we need to work individually.

(-8*-9) -> 2 (For two position of -8*-9 or -9*-8) * 2 (+/- x) * 2 (+/- y) = 8
(-12,-6) -> 2 (For two position of -8*-9 or -9*-8) * 2 (+/- x) * 2 (+/- y) = 8
(-2,-36) -> 1 ( only one way since -36 is only possible for the right term and not the left one) * 2 (+/- x) * 2 (+/- y) = 4
(-4,-18) -> 1 (only one way since -18 is not possible for the left side term but only the right one) * 2 (+/- x) * 2 (+/- y) = 4
(-3,-24) -> 1 (only one way since -24 is not possible for the left side term but only the right one) * 2 (+/- x) * 2 (+/- y) = 4

Sum = 16+12 = 28

Total = 48 + 28 =   76 . Off by 4

Now where I was wrong is that for the negative cases for (-12,-6) and (-2,-36), -12 and -36 are in the terms and get cancelled out giving one one solution, ie.

-12,-6 : Case1: 1 (for arrangment -12,-6) * 1 (Only one value of x = 2) * 2( two solutions for y)
 Case2: 1 (for arrangment -6, -12) * 2 (Now x has two values) * 2 (two for y)
 Total ways = 6

Same issue for (-2,-36) : 1 (only one case for -36) * 2 (two values for x) * 1 (Only one value for y = -4)
 No. of ways = 2

Overcount of 4. So total is  72

Aarya_San · Answer

ok so here was my attempt, maybe I'm missing something:

72 -> {(8,9),(3,24),(6,12),(36,2),(18,4),(72,1)}

There are 6 pairs. Now these pairs can be interchanged eg: 8*9 or 9*8, therefore multiply 6 with 2 = 12 ways

Now for each case, x can be positive or negative, eg: let's say for (-8*-9) -> x = 6 or -2 -> (|6-2| - 12) = -8 or (|-2-2| -12) = -8. (Note: this idea will work for each pair that we have). So multiply 12 by 2 = 24 ways

Now the exact same scenario will play out for y as well: Multiple 24*2 = 48.

Most would think the question is done by now, but unfortunately we still have the negative numbers left.

Like the example I took above, 8*9 and 9*8 can give us 72 but -8*-9 or -9*-8 can also give us 72. So let's multiply 48 by 2 = 96  (This is a mistake!)

This problem with this approach is that we over look the fact that the equation is set up in such a way that some negative numbers aren't possible. So we need to work individually.

(-8*-9) -> 2 (For two position of -8*-9 or -9*-8) * 2 (+/- x) * 2 (+/- y) = 8
(-12,-6) -> 2 (For two position of -8*-9 or -9*-8) * 2 (+/- x) * 2 (+/- y) = 8
(-2,-36) -> 1 ( only one way since -36 is only possible for the right term and not the left one) * 2 (+/- x) * 2 (+/- y) = 4
(-4,-18) -> 1 (only one way since -18 is not possible for the left side term but only the right one) * 2 (+/- x) * 2 (+/- y) = 4
(-3,-24) -> 1 (only one way since -24 is not possible for the left side term but only the right one) * 2 (+/- x) * 2 (+/- y) = 4

Sum = 16+12 = 28

Total = 48 + 28 =   76 . Off by 4

Now where I was wrong is that for the negative cases for (-12,-6) and (-2,-36), -12 and -36 are in the terms and get cancelled out giving one one solution, ie.

-12,-6 : Case1: 1 (for arrangment -12,-6) * 1 (Only one value of x = 2) * 2( two solutions for y)
 Case2: 1 (for arrangment -6, -12) * 2 (Now x has two values) * 2 (two for y)
 Total ways = 6

Same issue for (-2,-36) : 1 (only one case for -36) * 2 (two values for x) * 1 (Only one value for y = -4)
 No. of ways = 2

Overcount of 4. So total is  72

If x & y are integers, how many solution pairs (x,y) satisfy the equat

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If x & y are integers, how many solution pairs (x,y) satisfy the equat

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