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.
Here are some clues I found: I found that it is likely true as I
tested it for a wide range of values.
Not only is it true, but
if we consider a simpler problem for three integers,
where p + q + r = LCM(p,q,r), we find that all solutions are
divisible by 2 and 3, since all are a multiple of the 1st.
Not surprisingly, each 5th solution is divisible by 5.
p q r LCM 1/5? factored
-------------------------------------------------------------
1 2 3 6 **** 2^1 3^1
2 4 6 12 **** 2^2 3^1
3 6 9 18 **** 3^2 2^1
4 8 12 24 **** 2^3 3^1
5 10 15 30 5^1 2^1 3^1
6 12 18 36 **** 2^2 3^2
7 14 21 42 **** 7^1 2^1 3^1
8 16 24 48 **** 2^4 3^1
9 18 27 54 **** 3^3 2^1
10 20 30 60 2^2 5^1 3^1
11 22 33 66 **** 11^1 2^1 3^1
12 24 36 72 **** 2^3 3^2
13 26 39 78 **** 13^1 2^1 3^1
14 28 42 84 **** 2^2 7^1 3^1
15 30 45 90 3^2 5^1 2^1
16 32 48 96 **** 2^5 3^1
17 34 51 102 **** 17^1 2^1 3^1
18 36 54 108 **** 2^2 3^3
19 38 57 114 **** 19^1 2^1 3^1
20 40 60 120 2^3 5^1 3^1
21 42 63 126 **** 3^2 7^1 2^1
22 44 66 132 **** 2^2 11^1 3^1
23 46 69 138 **** 23^1 2^1 3^1
24 48 72 144 **** 2^4 3^2
25 50 75 150 5^2 2^1 3^1
26 52 78 156 **** 2^2 13^1 3^1
27 54 81 162 **** 3^4 2^1
28 56 84 168 **** 2^3 7^1 3^1
29 58 87 174 **** 29^1 2^1 3^1
30 60 90 180 2^2 3^2 5^1
Edited on February 5, 2023, 3:17 am
some clues
