Question 1093148
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That is wrong for 890 does not have three consecutive digits.
He should have ruled that case out.

There are two possible cases for the 3 digit number n, with three
consecutive digits.

Case 1:

hundreds digit = h
tens digit = h+1
hundreds digit = h+2

100h + 10(h+1) + h+2 = 100h + 10h + 10 + h+2 = 111h + 12   

This must be equal to the sum of 4 multiples of 10.
The sum of 4 multiples of 10 is itself a multiple of 10.
Suppose A is such that this multiple of 10 is 10A.  Then

111h + 12 = 10A

Write each number in terms of its nearest multiple of 10

(110+1)h + 10+2 = 10A

110h + h + 10 + 2 = 10A

Divide through by 10

11h + h/10 + 1 + 2/10 = A

h/10 + 2/10 = A - 11h - 2 

The right side is an integer so the left side must
also be an integer, say B.

h/10 + 2/10 = B 

h + 2 = 10B
    h = 10B - 2

Since h is a digit, 0 through 9, then B can only be 1
    h = 10(1)-2 = 10-2 = 8

But if h=8, then the tens digit is 9 and the hundreds
digit would be 10, but 10 is not a digit.  So case 1
is eliminated.

The other tutor should have eliminated this case.


So we can only consider case 2.

Case 2:

ones digit = u
tens digit = u+1
hundreds digit = u+2

100(u+2) + 10(u+1) + u = 100u + 200 + 10u + 10 + u = 111u + 210 

This must be equal to the sum of 4 multiples of 10.
The sum of 4 multiples of 10 is itself a multiple of 10.
Suppose A is such that this multiple of 10 is 10A.  
Then the sum of the 4 multiples of 10 is 10A

111u + 210 = 10A

Write 111 in terms of its nearest multiple of 10, which is 110.

(110+1)u + 210 = 10A

110u + u + 210 = 10A

Divide through by 10

11u + u/10 + 21 = A

u/10 = A - 11u - 21

Since the right side is an integer, so is the
left side.  Suppose that integer is B, then

u/10 = B

u = 10B

But u is a digit, and the only digit that is
a multiple of 10 is 0, so u = 0

So

ones digit = u = 0
tens digit = u+1 = 1
hundreds digit = u+2 = 2

So the 3 digit number is 210.   <-- the only interesting answer!
                                    Too bad you weren't asked that. 

But sadly, you were asked what are the 4 multiples of 10.

[Really! They should have asked you what is the 3-digit number. 
That's the interesting answer! For the 3 digit number must 
be 210 regardless of what the four multiples of 10 are]

But let's find what you were asked for, uninteresting as it is.

The sum of those 4 multiples of 10 is 10A,
so we substitute u=0

  111u + 210 = 10A

111(0) + 210 = 10A

         210 = 10A

So any 4 multiples of 10 that have sum 210 will
be an answer to the question.

The answer is any of these 120 groups of 4 multiples 
of 10 that have sum 210.  They are all listed below.
Be sure to tell your teacher that it would have 
been a million times more interesting to have 
asked the question "What is the 3 digit number?", 
the answer which would have been 210 regardless 
of which of the 120 groups of 4 multiples of 10 
could have been chosen.  But here are all 120
answers:

1.  {0,0,0,210}
2.  {0,0,10,200}
3.  {0,0,20,190}
4.  {0,0,30,180}
5.  {0,0,40,170}
6.  {0,0,50,160}
7.  {0,0,60,150}
8.  {0,0,70,140}
9.  {0,0,80,130}
10.  {0,0,90,120}
11.  {0,0,100,110}
12.  {0,10,10,190}
13.  {0,10,20,180}
14.  {0,10,30,170}
15.  {0,10,40,160}
16.  {0,10,50,150}
17.  {0,10,60,140}
18.  {0,10,70,130}
19.  {0,10,80,120}
20.  {0,10,90,110}
21.  {0,10,100,100}
22.  {0,20,20,170}
23.  {0,20,30,160}
24.  {0,20,40,150}
25.  {0,20,50,140}
26.  {0,20,60,130}
27.  {0,20,70,120}
28.  {0,20,80,110}
29.  {0,20,90,100}
30.  {0,30,30,150}
31.  {0,30,40,140}
32.  {0,30,50,130}
33.  {0,30,60,120}
34.  {0,30,70,110}
35.  {0,30,80,100}
36.  {0,30,90,90}
37.  {0,40,40,130}
38.  {0,40,50,120}
39.  {0,40,60,110}
40.  {0,40,70,100}
41.  {0,40,80,90}
42.  {0,50,50,110}
43.  {0,50,60,100}
44.  {0,50,70,90}
45.  {0,50,80,80}
46.  {0,60,60,90}
47.  {0,60,70,80}
48.  {0,70,70,70}
49.  {10,10,10,180}
50.  {10,10,20,170}
51.  {10,10,30,160}
52.  {10,10,40,150}
53.  {10,10,50,140}
54.  {10,10,60,130}
55.  {10,10,70,120}
56.  {10,10,80,110}
57.  {10,10,90,100}
58.  {10,20,20,160}
59.  {10,20,30,150}
60.  {10,20,40,140}
61.  {10,20,50,130}
62.  {10,20,60,120}
63.  {10,20,70,110}
64.  {10,20,80,100}
65.  {10,20,90,90}
66.  {10,30,30,140}
67.  {10,30,40,130}
68.  {10,30,50,120}
69.  {10,30,60,110}
70.  {10,30,70,100}
71.  {10,30,80,90}
72.  {10,40,40,120}
73.  {10,40,50,110}
74.  {10,40,60,100}
75.  {10,40,70,90}
76.  {10,40,80,80}
77.  {10,50,50,100}
78.  {10,50,60,90}
79.  {10,50,70,80}
80.  {10,60,60,80}
81.  {10,60,70,70}
82.  {20,20,20,150}
83.  {20,20,30,140}
84.  {20,20,40,130}
85.  {20,20,50,120}
86.  {20,20,60,110}
87.  {20,20,70,100}
88.  {20,20,80,90}
89.  {20,30,30,130}
90.  {20,30,40,120}
91.  {20,30,50,110}
92.  {20,30,60,100}
93.  {20,30,70,90}
94.  {20,30,80,80}
95.  {20,40,40,110}
96.  {20,40,50,100}
97.  {20,40,60,90}
98.  {20,40,70,80}
99.  {20,50,50,90}
100.  {20,50,60,80}
101.  {20,50,70,70}
102.  {20,60,60,70}
103.  {30,30,30,120}
104.  {30,30,40,110}
105.  {30,30,50,100}
106.  {30,30,60,90}
107.  {30,30,70,80}
108.  {30,40,40,100}
109.  {30,40,50,90}
110.  {30,40,60,80}
111.  {30,40,70,70}
112.  {30,50,50,80}
113.  {30,50,60,70}
114.  {30,60,60,60}
115.  {40,40,40,90}
116.  {40,40,50,80}
117.  {40,40,60,70}
118.  {40,50,50,70}
119.  {40,50,60,60}
120.  {50,50,50,60}

Edwin</pre></font></b>