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# Math problem

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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
+2558
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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
+124701
+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

Apr 15, 2022
edited by CPhill  Apr 15, 2022