The least common multiple (LCM) of four positive integers p, q, r, and s is equal to p+q+r+s
Is p*q*r*s always divisible by at least one of 3 and 5?
- If so, prove it.
- If not, provide a counterexample.
First, it is plain as to why multiples of one solution are also solutions:
the sum of p+q+r... is multiplied by n and the LCM is multiplied by n to
make a new solution, where n is an integer. All n work.
The outstanding math question is how new solutions occur or stop occurring.
For example, for the case of two integers, p+q, I think there are _no_ solutions
to p+q = LCM(p,q)
What happens with 5 integers? Here, new solutions keep occurring amidst the multiples
of the old ones, although I have not explored the reaches of the space well...
And with 5 integers: p, q, r, s, t, the prime factor 7 can now appear solo,
without 3 and 5, that is.
(a longer list is here which includes unique solution LCM=403) (a list of "firsts"
where we note that the primes 17 and 23 can participate)
lord@rabbit 12844 % fac5
ind# multiple p q r s t LCM solo 7 factors
---------------------------------------------------------------
1 ( 1x # 1) 1 1 1 2 5 10 2^1 5^1
2 ( 1x # 2) 1 1 1 6 9 18 2^1 3^2
3 ( 1x # 3) 1 1 2 8 12 24 2^3 3^1
4 ( 1x # 4) 1 1 3 3 4 12 3^1 2^2
5 ( 1x # 5) 1 1 3 5 5 15 3^1 5^1
6 ( 1x # 6) 1 1 3 10 15 30 3^1 2^1 5^1
7 ( 1x # 7) 1 1 4 4 10 20 2^2 5^1
8 ( 1x # 8) 1 1 4 12 18 36 2^2 3^2
9 ( 1x # 9) 1 1 6 8 8 24 2^3 3^1
10 ( 1x #10) 1 1 6 16 24 48 2^4 3^1
11 ( 1x #11) 1 1 8 10 20 40 2^3 5^1
12 ( 1x #12) 1 2 2 2 7 14 **** 2^1 7^1
13 ( 1x #13) 1 2 2 3 4 12 2^2 3^1
14 ( 1x #14) 1 2 2 10 15 30 2^1 5^1 3^1
15 ( 1x #15) 1 2 3 3 9 18 2^1 3^2
16 ( 1x #16) 1 2 3 12 18 36 2^2 3^2
17 ( 1x #17) 1 2 4 7 14 28 **** 2^2 7^1
18 ( 1x #18) 1 2 6 6 15 30 2^1 3^1 5^1
19 ( 1x #19) 1 2 9 12 12 36 2^2 3^2
20 ( 1x #20) 1 3 3 3 5 15 3^1 5^1
21 ( 1x #21) 1 3 3 7 7 21 3^1 7^1
22 ( 1x #22) 1 3 3 14 21 42 3^1 2^1 7^1
23 ( 1x #23) 1 3 4 8 8 24 3^1 2^3
24 ( 1x #24) 1 3 4 16 24 48 3^1 2^4
25 ( 1x #25) 1 3 5 6 15 30 3^1 5^1 2^1
26 ( 1x #26) 1 3 6 6 8 24 3^1 2^3
27 ( 1x #27) 1 3 6 10 10 30 3^1 2^1 5^1
28 ( 1x #28) 1 3 12 16 16 48 3^1 2^4
29 ( 1x #29) 1 4 4 9 18 36 2^2 3^2
30 ( 1x #30) 1 4 5 5 5 20 2^2 5^1
31 ( 1x #31) 1 4 15 20 20 60 2^2 3^1 5^1
32 ( 1x #32) 1 5 9 15 15 45 5^1 3^2
33 ( 1x #33) 1 6 7 7 21 42 2^1 3^1 7^1
34 ( 1x #34) 1 6 7 14 14 42 2^1 3^1 7^1
35 ( 1x #35) 1 12 12 15 20 60 2^2 3^1 5^1
36 ( 1x #36) 2 2 2 3 9 18 2^1 3^2
37 ( 2x # 1) 2 2 2 4 10 20 2^2 5^1
38 ( 2x # 2) 2 2 2 12 18 36 2^2 3^2
39 ( 1x #39) 2 2 3 14 21 42 2^1 3^1 7^1
40 ( 2x # 3) 2 2 4 16 24 48 2^4 3^1
41 ( 1x #41) 2 2 5 6 15 30 2^1 5^1 3^1
42 ( 2x # 4) 2 2 6 6 8 24 2^3 3^1
43 ( 2x # 5) 2 2 6 10 10 30 2^1 3^1 5^1
44 ( 2x # 7) 2 2 8 8 20 40 2^3 5^1
45 ( 2x # 9) 2 2 12 16 16 48 2^4 3^1
46 ( 1x #46) 2 3 3 8 8 24 2^3 3^1
47 ( 1x #47) 2 3 3 16 24 48 2^4 3^1
48 ( 1x #48) 2 3 4 9 18 36 2^2 3^2
49 ( 1x #49) 2 3 5 5 15 30 2^1 3^1 5^1
50 ( 1x #50) 2 3 5 10 10 30 2^1 3^1 5^1
51 ( 1x #51) 2 3 15 20 20 60 2^2 3^1 5^1
52 ( 2x #12) 2 4 4 4 14 28 **** 2^2 7^1
53 ( 1x #53) 2 4 4 5 5 20 2^2 5^1
54 ( 2x #13) 2 4 4 6 8 24 2^3 3^1
55 ( 2x #15) 2 4 6 6 18 36 2^2 3^2
56 ( 1x #56) 2 4 9 9 12 36 2^2 3^2
57 ( 2x #19) 2 4 18 24 24 72 2^3 3^2
58 ( 1x #58) 2 5 5 8 20 40 2^3 5^1
59 ( 2x #20) 2 6 6 6 10 30 2^1 3^1 5^1
60 ( 1x #60) 2 6 6 7 21 42 2^1 3^1 7^1
61 ( 2x #21) 2 6 6 14 14 42 2^1 3^1 7^1
62 ( 2x #23) 2 6 8 16 16 48 2^4 3^1
63 ( 2x #26) 2 6 12 12 16 48 2^4 3^1
64 ( 2x #27) 2 6 12 20 20 60 2^2 3^1 5^1
65 ( 2x #30) 2 8 10 10 10 40 2^3 5^1
66 ( 3x # 1) 3 3 3 6 15 30 3^1 2^1 5^1
67 ( 1x #67) 3 3 4 6 8 24 3^1 2^3
68 ( 3x # 4) 3 3 9 9 12 36 3^2 2^2
69 ( 3x # 5) 3 3 9 15 15 45 3^2 5^1
70 ( 3x # 9) 3 3 18 24 24 72 3^2 2^3
71 ( 1x #71) 3 5 6 6 10 30 3^1 5^1 2^1
72 ( 1x #72) 3 5 12 20 20 60 3^1 5^1 2^2
73 ( 3x #12) 3 6 6 6 21 42 3^1 2^1 7^1
74 ( 3x #13) 3 6 6 9 12 36 3^2 2^2
75 ( 3x #20) 3 9 9 9 15 45 3^2 5^1
76 ( 3x #21) 3 9 9 21 21 63 3^2 7^1
77 ( 3x #23) 3 9 12 24 24 72 3^2 2^3
78 ( 3x #26) 3 9 18 18 24 72 3^2 2^3
79 ( 1x #79) 3 10 12 15 20 60 3^1 2^2 5^1
80 ( 3x #30) 3 12 15 15 15 60 3^1 2^2 5^1
81 ( 2x #36) 4 4 4 6 18 36 2^2 3^2
82 ( 4x # 1) 4 4 4 8 20 40 2^3 5^1
83 ( 4x # 4) 4 4 12 12 16 48 2^4 3^1
84 ( 4x # 5) 4 4 12 20 20 60 2^2 3^1 5^1
85 ( 2x #46) 4 6 6 16 16 48 2^4 3^1
86 ( 2x #50) 4 6 10 20 20 60 2^2 3^1 5^1
87 ( 1x #87) 4 6 15 15 20 60 2^2 3^1 5^1
88 ( 2x #53) 4 8 8 10 10 40 2^3 5^1
89 ( 4x #13) 4 8 8 12 16 48 2^4 3^1
90 ( 2x #56) 4 8 18 18 24 72 2^3 3^2
91 ( 4x #20) 4 12 12 12 20 60 2^2 3^1 5^1
92 ( 4x #30) 4 16 20 20 20 80 2^4 5^1
93 ( 5x # 4) 5 5 15 15 20 60 5^1 3^1 2^2
94 ( 5x #13) 5 10 10 15 20 60 5^1 2^2 3^1
95 ( 2x #67) 6 6 8 12 16 48 2^4 3^1
96 ( 6x # 4) 6 6 18 18 24 72 2^3 3^2
97 ( 3x #46) 6 9 9 24 24 72 2^3 3^2
98 ( 2x #71) 6 10 12 12 20 60 2^2 3^1 5^1
99 ( 3x #53) 6 12 12 15 15 60 2^2 3^1 5^1
100 ( 6x #13) 6 12 12 18 24 72 2^3 3^2
101 ( 4x #53) 8 16 16 20 20 80 2^4 5^1
102 ( 3x #67) 9 9 12 18 24 72 3^2 2^3
Edited on February 16, 2023, 6:33 am
with five
