## A student at Harvard named Kevin Was counting his stones by 11 He messed up n times And instead counted 9s And wound

Question

A student at Harvard named Kevin
Was counting his stones by 11
He messed up n times
And wound up at 2007.

To be precise, Kevin divides 2007 stones into groups of 9 stones and 11 stones, where n of the groups have 9 stones in them. How many values of could make this limerick true?

(Kevin can mess up every time)

in progress 0
2 weeks 2021-09-15T02:22:25+00:00 1 Answer 0

Work:
Some groups of 11,
and some groups of 9,
together make 2007.

We may make an equation
11x + 9y = 2007,
and we need to find how many possibilities there are for y.

2007/9=223
So, the highest value for y is 223.

Let k be a value for y.
If k is a valid value for y then (2007-9k) must be divisible by 11.

Let’s just brute force this by doing this validity test with every number from 1 to 223. Then we can count how many valid options there are.

0 is NOT valid
1 is valid
2 is NOT valid
3 is NOT valid
4 is NOT valid
5 is NOT valid
6 is NOT valid
7 is NOT valid
8 is NOT valid
9 is NOT valid
10 is NOT valid
11 is NOT valid
12 is valid
13 is NOT valid
14 is NOT valid
15 is NOT valid
16 is NOT valid
17 is NOT valid
18 is NOT valid
19 is NOT valid
20 is NOT valid
21 is NOT valid
22 is NOT valid
23 is valid
24 is NOT valid
25 is NOT valid
26 is NOT valid
27 is NOT valid
28 is NOT valid
29 is NOT valid
30 is NOT valid
31 is NOT valid
32 is NOT valid
33 is NOT valid
34 is valid
35 is NOT valid
36 is NOT valid
37 is NOT valid
38 is NOT valid
39 is NOT valid
40 is NOT valid
41 is NOT valid
42 is NOT valid
43 is NOT valid
44 is NOT valid
45 is valid
46 is NOT valid
47 is NOT valid
48 is NOT valid
49 is NOT valid
50 is NOT valid
51 is NOT valid
52 is NOT valid
53 is NOT valid
54 is NOT valid
55 is NOT valid
56 is valid
57 is NOT valid
58 is NOT valid
59 is NOT valid
60 is NOT valid
61 is NOT valid
62 is NOT valid
63 is NOT valid
64 is NOT valid
65 is NOT valid
66 is NOT valid
67 is valid
68 is NOT valid
69 is NOT valid
70 is NOT valid
71 is NOT valid
72 is NOT valid
73 is NOT valid
74 is NOT valid
75 is NOT valid
76 is NOT valid
77 is NOT valid
78 is valid
79 is NOT valid
80 is NOT valid
81 is NOT valid
82 is NOT valid
83 is NOT valid
84 is NOT valid
85 is NOT valid
86 is NOT valid
87 is NOT valid
88 is NOT valid
89 is valid
90 is NOT valid
91 is NOT valid
92 is NOT valid
93 is NOT valid
94 is NOT valid
95 is NOT valid
96 is NOT valid
97 is NOT valid
98 is NOT valid
99 is NOT valid
100 is valid
101 is NOT valid
102 is NOT valid
103 is NOT valid
104 is NOT valid
105 is NOT valid
106 is NOT valid
107 is NOT valid
108 is NOT valid
109 is NOT valid
110 is NOT valid
111 is valid
112 is NOT valid
113 is NOT valid
114 is NOT valid
115 is NOT valid
116 is NOT valid
117 is NOT valid
118 is NOT valid
119 is NOT valid
120 is NOT valid
121 is NOT valid
122 is valid
123 is NOT valid
124 is NOT valid
125 is NOT valid
126 is NOT valid
127 is NOT valid
128 is NOT valid
129 is NOT valid
130 is NOT valid
131 is NOT valid
132 is NOT valid
133 is valid
134 is NOT valid
135 is NOT valid
136 is NOT valid
137 is NOT valid
138 is NOT valid
139 is NOT valid
140 is NOT valid
141 is NOT valid
142 is NOT valid
143 is NOT valid
144 is valid
145 is NOT valid
146 is NOT valid
147 is NOT valid
148 is NOT valid
149 is NOT valid
150 is NOT valid
151 is NOT valid
152 is NOT valid
153 is NOT valid
154 is NOT valid
155 is valid
156 is NOT valid
157 is NOT valid
158 is NOT valid
159 is NOT valid
160 is NOT valid
161 is NOT valid
162 is NOT valid
163 is NOT valid
164 is NOT valid
165 is NOT valid
166 is valid
167 is NOT valid
168 is NOT valid
169 is NOT valid
170 is NOT valid
171 is NOT valid
172 is NOT valid
173 is NOT valid
174 is NOT valid
175 is NOT valid
176 is NOT valid
177 is valid
178 is NOT valid
179 is NOT valid
180 is NOT valid
181 is NOT valid
182 is NOT valid
183 is NOT valid
184 is NOT valid
185 is NOT valid
186 is NOT valid
187 is NOT valid
188 is valid
189 is NOT valid
190 is NOT valid
191 is NOT valid
192 is NOT valid
193 is NOT valid
194 is NOT valid
195 is NOT valid
196 is NOT valid
197 is NOT valid
198 is NOT valid
199 is valid
200 is NOT valid
201 is NOT valid
202 is NOT valid
203 is NOT valid
204 is NOT valid
205 is NOT valid
206 is NOT valid
207 is NOT valid
208 is NOT valid
209 is NOT valid
210 is valid
211 is NOT valid
212 is NOT valid
213 is NOT valid
214 is NOT valid
215 is NOT valid
216 is NOT valid
217 is NOT valid
218 is NOT valid
219 is NOT valid
220 is NOT valid
221 is valid
222 is NOT valid
223 is NOT valid

21 valid options for y!