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
