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Four distinct integers a, b, c and d have the property that when added in pairs, the sums 10, 18, 19, 22, 23, 31 are obtained. What are the four integers in increasing order? (place a comma and then a space between each integer)

 Apr 15, 2022
 #1
avatar+1384 
+1

We have 6 equations: 

 

\(a+b=10\)    (1)

\(a+c=18\)    (2)

\(a+d=19\)    (3)

\(b+c=22\)     (4)

\(b+d=23\)    (5)

\(c+d=31\)    (6)

 

Using equations 2 and 3, we see that \(d = 1+c\)

 

Subbing this into (6), we get: \(c+c+1=31\), meaning \(c=15\)

 

Subbing this into (2), we find \(a=3\)

 

Subbing the value of c into (4), we find \(b=7\)

 

Subbing the value of c into (6), we find \(d = 16\)

 

Thus, the values are \(\color{brown}\boxed{3,7,15,16}\)

 Apr 15, 2022
 #2
avatar+122390 
+1

Let    a,b, c, d   be in increasing order

 

Using some logic....we know that 

a + b  = 10       (1)

a + c =  18       (2)

c + d =  31      (3)

b + d =  23      (4)

 

What we don't know is  that if

 

a + d =  19   or   a + d  =  22  or   b + c = 19  or   b + c =   22

 

To see which is true    subtract   (4)  from (1)    and we  get 

 

a - d  =  -13        (5)

 

Now let us assume that   a + d  =  22      (6)

 

Add (5) and (6)  and we get that

 

2a = 9             but a here is not an integer

 

So....it must be that

 

a + d = 19

 

So

 

a - d = -13

a + d = 19          add these

 

2a  = 6

 

So.....using our equations.....

 

a = 3

b = 7

c = 15

d = 16

 

 

cool cool cool

 Apr 15, 2022
edited by CPhill  Apr 15, 2022

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