Ah yes, thanks to all of you, of course, c divides a*b, and from there your reasoning, quite clear, it matches the solutions found, 2, 29, 2 3, 13, 13, 5, 11, 11 best regards and thanks. simon plouffe Le 2020-10-02 à 18:45, Tomas Rokicki a écrit :
For the prime case I'd do this.
Rewrite
a*b + 7*a*c + 15*c = a*b*c
as
a b (c - 1) == c (7a + 15)
If c==2 then this simplifies to
b = 14 + 30/aj
which (going through the prime divisors of 30, which number 3) gives us the solution (a=2, b=29, c=2).
So other than this solution c>2. The left side must be divisible by c, but c-1 never is divisible by the prime c, so either a=c or b=c. We handle the two cases in turn.
If a=c then we have
b = 7 + 22 / (c - 1)
So for b to be integral c must be 1, 2, 11, or 22. None of these give a new solution.
If b=c, then we have:
c = 8 + 15/a
a can be 3 or 5, and this gives c=13, 11 respectively so this gives the solutions (a=13, b=3, c=3) and (a=11, b=5, c=5).
-tom
On Fri, Oct 2, 2020 at 8:49 AM Simon Plouffe <simon.plouffe@gmail.com> wrote:
Here is a simple problem I saw in one math marathon at the University of Orsay for youngsters, it is similar to the PUTNAM.
If we have a*b + 7*a*c + 15*c = a*b*c then find all a,b,c that are primes that satisfies the equation. a,b,c are all >=2.
By using brute force I can find 3 solution with small primes and 49 others with non-primes, 52 solutions in all.
I am stuck with this problem, I do not see how to resolve this without using brute force.
Normally, that problem should be feasible with simple means, I don't see how to find the 52 solutions and from that , the 3 triplets of primes.
hint, a, b c are relatively small one of the solution is 2, 29, 2 for a, b, c,
see below for the complete set (spoiler). here is the original paper : https://www.imo.universite-paris-saclay.fr/marathon/sept20.pdf
Best regards, Simon Plouffe
1, 23, 23 1, 24, 12 1, 33, 3 1, 44, 2 2, 15, 30 2, 29, 2 3, 13, 13 3, 14, 7 3, 15, 5 3, 16, 4 3, 18, 3 3, 24, 2 4, 11, 44 5, 11, 11 5, 12, 6 5, 15, 3 5, 20, 2 6, 10, 20 6, 19, 2 8, 9, 72 9, 9, 27 9, 13, 3 10, 9, 18 10, 17, 2 12, 9, 12 12, 11, 4 15, 9, 9 15, 10, 5 15, 12, 3 15, 16, 2 16, 8, 128 17, 8, 68 18, 8, 48 19, 8, 38 20, 8, 32 21, 8, 28 21, 9, 7 23, 8, 23 25, 8, 20 27, 8, 18 30, 8, 16 30, 9, 6 30, 10, 4 30, 15, 2 35, 8, 14 39, 8, 13 45, 8, 12 45, 11, 3 55, 8, 11 75, 8, 10 75, 9, 5 135, 8, 9
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