20070708, 16:16  #1 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Always an integer.
Show that (x^5)/5 + (x^3)/3 +(7x/15) is always an integer for integral values of x ? Its easy if you know the method. Similar problems are welcome. Mally 
20070708, 17:24  #2 
Jun 2007
Moscow,Russia
7·19 Posts 
Well, I don't know the simplest way to reduce this problem, but it can be easy solved with one qiute obviously notice: any integer value can be represented as element of the set {15*k,15*k+1,15*k+2,...,15*k+14} for integer k=0,1,... Thus we can substitute this values for x in out initial expression and get results . Mathematica code, returns values of initial expression by substituting for x:
Code:
(1/15) x (7 + 5 x^2 + 3 x^4) /. x > 15*k + Range[0, 14] // Expand Code:
{7 k + 1125 k^3 + 151875 k^5, 1 + 37 k + 675 k^2 + 7875 k^3 + 50625 k^4 + 151875 k^5, 10 + 307 k + 4050 k^2 + 28125 k^3 + 101250 k^4 + 151875 k^5, 59 + 1357 k + 12825 k^2 + 61875 k^3 + 151875 k^4 + 151875 k^5, 228 + 4087 k + 29700 k^2 + 109125 k^3 + 202500 k^4 + 151875 k^5, 669 + 9757 k + 57375 k^2 + 169875 k^3 + 253125 k^4 + 151875 k^5, 1630 + 19987 k + 98550 k^2 + 244125 k^3 + 303750 k^4 + 151875 k^5, 3479 + 36757 k + 155925 k^2 + 331875 k^3 + 354375 k^4 + 151875 k^5, 6728 + 62407 k + 232200 k^2 + 433125 k^3 + 405000 k^4 + 151875 k^5, 12057 + 99637 k + 330075 k^2 + 547875 k^3 + 455625 k^4 + 151875 k^5, 20338 + 151507 k + 452250 k^2 + 676125 k^3 + 506250 k^4 + 151875 k^5, 32659 + 221437 k + 601425 k^2 + 817875 k^3 + 556875 k^4 + 151875 k^5, 50348 + 313207 k + 780300 k^2 + 973125 k^3 + 607500 k^4 + 151875 k^5, 74997 + 430957 k + 991575 k^2 + 1141875 k^3 + 658125 k^4 + 151875 k^5, 108486 + 579187 k + 1237950 k^2 + 1324125 k^3 + 708750 k^4 + 151875 k^5} Last fiddled with by VolMike on 20070708 at 17:40 
20070708, 17:27  #3  
Jun 2003
2^{2}×3×439 Posts 
Quote:
Alternately we need to show that 3x^5 + 5x^3 + 7x is a multiple of 15. Working modulo 3, we have x^3=x (by Fermat's Little Theorem) Thus 3x^5 + 5x^3 + 7x == 3x^5 + 5x + 7x == 3x^5 + 12x == 0 (mod 3) Working modulo 5, we have x^5=x Thus 3x^5 + 5x^3 + 7x == 3x + 5x^3 + 7x == 10x + 5x^3 == 0 (mod 5) Thus our expression is divisible by 15, as required. Q.E.D 

20070708, 18:35  #4 
"Lucan"
Dec 2006
England
194A_{16} Posts 
I tried to show 3x^5+5x^3+7x was a multiple of 15
by induction and failed 
20070708, 19:14  #5 
(loop (#_fork))
Feb 2006
Cambridge, England
2^{2}·1,613 Posts 
Proofs by working to a single modulus M are, surely, precisely proofs by induction; just with M distinct base cases and the induction rule being n > n+M.
But in fact it seems to work for me by straight n>n+1 induction: 3(x+1)^5 = 3x^5 + 15x^4+30x^3+30x^2+15x+3 5(x+1)^3 = 5x^3 + 15x^2 + 15x + 5 7(x+1) = 7x + 7 so the sum is 3x^5 + 5x^3 + 7x (ex hypothesi divisible by 15) + lots of things which are in form a multiple of 15 + 3 + 5 + 7 which equals 15 
20070708, 19:50  #6 
"Lucan"
Dec 2006
England
14512_{8} Posts 
Thanks FiveMack.
Actually I tried to prove that 3x^4 +5x^2 +7 was divisible by 15. This could indeed be false, when x is a multiple of 3 and/or 5. David And THX Mally  a successful puzzle :) Look forward to the next! Last fiddled with by davieddy on 20070708 at 20:28 
20070709, 15:52  #7  
Bronze Medalist
Jan 2004
Mumbai,India
804_{16} Posts 
Precise and concise!
Quote:
Hold your horses Davie. Here is a precise and concise solution you can ever get. Thats why I put this problem. Its the type Alpertron and Maxal will appreciate so here it is. original Expression = [(x^5x)/5] + [(x^3x)/3] + x. But by Fermat's theorem x^5  x == 0 mod 5 and x^3 x ==Mod 3 hence the expression is an integer! Q.E.D Mally 

20070709, 19:23  #8 
"Lucan"
Dec 2006
England
14512_{8} Posts 

20070709, 23:49  #9 
Feb 2007
660_{8} Posts 
For me [(x^nx)/n] is an integer for any n and x...
since [.] means floor(.) to me... especially when it is put where it is not needed for anything else like grouping factors of a product or function arguments etc... 
20070709, 23:58  #10 
Feb 2007
1B0_{16} Posts 

20070712, 08:33  #11  
Bronze Medalist
Jan 2004
Mumbai,India
4004_{8} Posts 
General Eqn.
Quote:
Excellent Davie! I would like you to note that the general expression is true for all +ve integers when p is a prime It is also true if p divides x. This is a corollary which results from Fermat’s little theorem. Thus from your very expression when x is any number e.g. x = 14 14^7/7 + 14^2/2 +5*14/7 an integer although 7 and 2 both divide 14 A more restricted expression can be derived from Fermat’s little theorem, thus x^6/7 + x^2/3 – 10/21 is also true and always an integer provided the primes (In denominator) do not divide x , and the constant is changed accordingly as it’s not a function of x but rather of the primes. Having said that could you derive the general expression in terms of x, p_1, and p_2……? Try it with x = 5, 11 and any prime. Mally . 

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