Difference between revisions of "2005 JBMO Problems/Problem 4"

(Created page with "Let (a, b, c) = k Then a = pk b = qk c = rk 100a + 10b + c = abc(a + b + c) 100p + 10q + r = pqr(p + q + r)k^3 We see that if any of (a, b, c) is 0, all the terms are 0, an...")
 
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Then
 
Then
 
a = pk
 
a = pk
 +
 
b = qk  
 
b = qk  
 +
 
c = rk
 
c = rk
  
 
100a + 10b + c = abc(a + b + c)
 
100a + 10b + c = abc(a + b + c)
 +
 
100p + 10q + r = pqr(p + q + r)k^3
 
100p + 10q + r = pqr(p + q + r)k^3
  
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Let (p, q) = c with p = cs and q = cy
 
Let (p, q) = c with p = cs and q = cy
 +
 
Then
 
Then
     c(100s + 10y) + r = (c^2)syr(cs + cu + r)k^3
+
      
 +
    c(100s + 10y) + r = (c^2)syr(cs + cu + r)k^3
 +
 
 
We see that c is also a factor of r.
 
We see that c is also a factor of r.
  
 
We get similar results for (p, r) = c and (q, r) = c
 
We get similar results for (p, r) = c and (q, r) = c
 +
 
The terms cancel, and thus we get that
 
The terms cancel, and thus we get that
 +
 
For some (m, n, t, j) which are respectively factors of (a, b, c)
 
For some (m, n, t, j) which are respectively factors of (a, b, c)
  
 
     100m + 10n + t = mnt(m + n + t)j^3
 
     100m + 10n + t = mnt(m + n + t)j^3
 +
 
With  0 < m, n, t, j < 10
 
With  0 < m, n, t, j < 10
  
 
The computation is left to the reader
 
The computation is left to the reader

Revision as of 11:58, 11 April 2019

Let (a, b, c) = k Then a = pk

b = qk

c = rk

100a + 10b + c = abc(a + b + c)

100p + 10q + r = pqr(p + q + r)k^3

We see that if any of (a, b, c) is 0, all the terms are 0, and such (a, b, c) are nonzero

We will show that we can find (m, n, t) such that m, n, and t are factors of a, b, and c respectively, with (m, n, p) being pairwise relatively prime.

Let (p, q) = c with p = cs and q = cy

Then

    c(100s + 10y) + r = (c^2)syr(cs + cu + r)k^3

We see that c is also a factor of r.

We get similar results for (p, r) = c and (q, r) = c

The terms cancel, and thus we get that

For some (m, n, t, j) which are respectively factors of (a, b, c)

    100m + 10n + t = mnt(m + n + t)j^3

With 0 < m, n, t, j < 10

The computation is left to the reader