Katie had a
collection of red, green and blue beads. She noticed that the number of beads
of each colour was a prime number and that the numbers were all different.
She also observed that if she multiplied the number of red beads by the total number of
red and green beads she obtained a number exactly 120 greater than the number
of blue beads.
How many beads of each colour did she have?
SOURCE: Scottish Mathematical Challenge.
(Sorry, I mixed 120 greater for 120 times greater). But it is also interesting, so I leave it.
Following precedent notation
R*(R+G)=120*B=2^3*3*5*B
R+G has to have as divisors all prime factors of 120 except prime factor R and also prime factor B (Because both sides of the equation should have the same prime factors).
If R=2 then R+G = 2^2*3*5*B = 60*B. Then G= 60*B - 2 which is a par number. So not prime.
If R= 3 then R+G = 2^3*5*B = 40*B. Then G = 40*B - 3
For B=5 G=197 which is prime. So one solution is
(R,G,B)=(3,197,5)
For B=7 G=177 not prime
For B=11 G=437 from here in my opinion there are too much beads to make sense.
If R=5 then R+G =2^3*3*B= 24*C Then G= 24*B-5
For B=3 G=67 which is prime
(R,G,B)=(5,67,3)
Edited on March 21, 2018, 10:53 am
Edited on March 21, 2018, 11:01 am
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Posted by armando
on 2018-03-21 10:45:42 |
Some solutions
