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# Counting

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How many ordered pairs of positive integers (m,n) satisfy lcm[m,n]=\(7200\)?

Feb 15, 2022

#1
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There are many pairs that satisfy your condition. Note that a pair such as (36, 800) is considred different from (800, 36) and therefore is counted twice as follows:

1 = (1, 7200)
2 = (2, 7200)
3 = (3, 7200)
4 = (4, 7200)
5 = (5, 7200)
6 = (6, 7200)
7 = (8, 7200)
8 = (9, 800)
9 = (9, 2400)
10 = (9, 7200)
11 = (10, 7200)
12 = (12, 7200)
13 = (15, 7200)
14 = (16, 7200)
15 = (18, 800)
16 = (18, 2400)
17 = (18, 7200)
18 = (20, 7200)
19 = (24, 7200)
20 = (25, 288)
21 = (25, 1440)
22 = (25, 7200)
23 = (30, 7200)
24 = (32, 225)
25 = (32, 450)
26 = (32, 900)
27 = (32, 1800)
28 = (32, 3600)
29 = (32, 7200)
30 = (36, 800)
31 = (36, 2400)
32 = (36, 7200)
33 = (40, 7200)
34 = (45, 800)
35 = (45, 2400)
36 = (45, 7200)
37 = (48, 7200)
38 = (50, 288)
39 = (50, 1440)
40 = (50, 7200)
41 = (60, 7200)
42 = (72, 800)
43 = (72, 2400)
44 = (72, 7200)
45 = (75, 288)
46 = (75, 1440)
47 = (75, 7200)
48 = (80, 7200)
49 = (90, 800)
50 = (90, 2400)
51 = (90, 7200)
52 = (96, 225)
53 = (96, 450)
54 = (96, 900)
55 = (96, 1800)
56 = (96, 3600)
57 = (96, 7200)
58 = (100, 288)
59 = (100, 1440)
60 = (100, 7200)
61 = (120, 7200)
62 = (144, 800)
63 = (144, 2400)
64 = (144, 7200)
65 = (150, 288)
66 = (150, 1440)
67 = (150, 7200)
68 = (160, 225)
69 = (160, 450)
70 = (160, 900)
71 = (160, 1800)
72 = (160, 3600)
73 = (160, 7200)
74 = (180, 800)
75 = (180, 2400)
76 = (180, 7200)
77 = (200, 288)
78 = (200, 1440)
79 = (200, 7200)
80 = (225, 32)
81 = (225, 96)
82 = (225, 160)
83 = (225, 288)
84 = (225, 480)
85 = (225, 800)
86 = (225, 1440)
87 = (225, 2400)
88 = (225, 7200)
89 = (240, 7200)
90 = (288, 25)
91 = (288, 50)
92 = (288, 75)
93 = (288, 100)
94 = (288, 150)
95 = (288, 200)
96 = (288, 225)
97 = (288, 300)
98 = (288, 400)
99 = (288, 450)
100 = (288, 600)
101 = (288, 800)
102 = (288, 900)
103 = (288, 1200)
104 = (288, 1800)
105 = (288, 2400)
106 = (288, 3600)
107 = (288, 7200)
108 = (300, 288)
109 = (300, 1440)
110 = (300, 7200)
111 = (360, 800)
112 = (360, 2400)
113 = (360, 7200)
114 = (400, 288)
115 = (400, 1440)
116 = (400, 7200)
117 = (450, 32)
118 = (450, 96)
119 = (450, 160)
120 = (450, 288)
121 = (450, 480)
122 = (450, 800)
123 = (450, 1440)
124 = (450, 2400)
125 = (450, 7200)
126 = (480, 225)
127 = (480, 450)
128 = (480, 900)
129 = (480, 1800)
130 = (480, 3600)
131 = (480, 7200)
132 = (600, 288)
133 = (600, 1440)
134 = (600, 7200)
135 = (720, 800)
136 = (720, 2400)
137 = (720, 7200)
138 = (800, 9)
139 = (800, 18)
140 = (800, 36)
141 = (800, 45)
142 = (800, 72)
143 = (800, 90)
144 = (800, 144)
145 = (800, 180)
146 = (800, 225)
147 = (800, 288)
148 = (800, 360)
149 = (800, 450)
150 = (800, 720)
151 = (800, 900)
152 = (800, 1440)
153 = (800, 1800)
154 = (800, 3600)
155 = (800, 7200)
156 = (900, 32)
157 = (900, 96)
158 = (900, 160)
159 = (900, 288)
160 = (900, 480)
161 = (900, 800)
162 = (900, 1440)
163 = (900, 2400)
164 = (900, 7200)
165 = (1200, 288)
166 = (1200, 1440)
167 = (1200, 7200)
168 = (1440, 25)
169 = (1440, 50)
170 = (1440, 75)
171 = (1440, 100)
172 = (1440, 150)
173 = (1440, 200)
174 = (1440, 225)
175 = (1440, 300)
176 = (1440, 400)
177 = (1440, 450)
178 = (1440, 600)
179 = (1440, 800)
180 = (1440, 900)
181 = (1440, 1200)
182 = (1440, 1800)
183 = (1440, 2400)
184 = (1440, 3600)
185 = (1440, 7200)
186 = (1800, 32)
187 = (1800, 96)
188 = (1800, 160)
189 = (1800, 288)
190 = (1800, 480)
191 = (1800, 800)
192 = (1800, 1440)
193 = (1800, 2400)
194 = (1800, 7200)
195 = (2400, 9)
196 = (2400, 18)
197 = (2400, 36)
198 = (2400, 45)
199 = (2400, 72)
200 = (2400, 90)
201 = (2400, 144)
202 = (2400, 180)
203 = (2400, 225)
204 = (2400, 288)
205 = (2400, 360)
206 = (2400, 450)
207 = (2400, 720)
208 = (2400, 900)
209 = (2400, 1440)
210 = (2400, 1800)
211 = (2400, 3600)
212 = (2400, 7200)
213 = (3600, 32)
214 = (3600, 96)
215 = (3600, 160)
216 = (3600, 288)
217 = (3600, 480)
218 = (3600, 800)
219 = (3600, 1440)
220 = (3600, 2400)
221 = (3600, 7200)
222 = (7200, 1)
223 = (7200, 2)
224 = (7200, 3)
225 = (7200, 4)
226 = (7200, 5)
227 = (7200, 6)
228 = (7200, 8)
229 = (7200, 9)
230 = (7200, 10)
231 = (7200, 12)
232 = (7200, 15)
233 = (7200, 16)
234 = (7200, 18)
235 = (7200, 20)
236 = (7200, 24)
237 = (7200, 25)
238 = (7200, 30)
239 = (7200, 32)
240 = (7200, 36)
241 = (7200, 40)
242 = (7200, 45)
243 = (7200, 48)
244 = (7200, 50)
245 = (7200, 60)
246 = (7200, 72)
247 = (7200, 75)
248 = (7200, 80)
249 = (7200, 90)
250 = (7200, 96)
251 = (7200, 100)
252 = (7200, 120)
253 = (7200, 144)
254 = (7200, 150)
255 = (7200, 160)
256 = (7200, 180)
257 = (7200, 200)
258 = (7200, 225)
259 = (7200, 240)
260 = (7200, 288)
261 = (7200, 300)
262 = (7200, 360)
263 = (7200, 400)
264 = (7200, 450)
265 = (7200, 480)
266 = (7200, 600)
267 = (7200, 720)
268 = (7200, 800)
269 = (7200, 900)
270 = (7200, 1200)
271 = (7200, 1440)
272 = (7200, 1800)
273 = (7200, 2400)
274 = (7200, 3600)
275 = (7200, 7200)
And there you have it! The answer is 275 such pairs.

Feb 16, 2022