Common multiple of positive integers

  • #1
100
0
Just have this question im having trouble with

The least common multiple of positive integers a, b, c and d is equal to a + b + c + d.
Prove that abcd is divisible by at least one of 3 and 5.

Thanks
 

Answers and Replies

  • #2
100
0
does anyone have an answer?
 
  • #4
92
0
man I'll give it a try but i'm not math pro so bear with me.

(1) lcm= a+b+c+d = x1a = x2b = x3c = x4d for x1-x4 are some integer.

raise to exponent 4 on the left side will yield : (a+b+c+d)^4=(x1a)(x2b)(x3c)(x4d)
rewrite it to : (a+b+c+d)^4 = x1x2x3x4(abcd)
divide by x1x2x3x4 : (a+b+c+d)^4/(x1x2x3x4)=abcd
divide both side by 15 which is 3 and 5 : (a+b+c+d)^4/(15x1x2x3x4)=abcd/15
now we want the left side to equal to some integer say k, let have k+1 for simplicity
k=1=(a+b+c+d)^4/(15x1x2x3x4)
rearange : x1x2x3x4=(a+b+c+d)^4/(15)
If we take a+b+c+d = 15, then x1x2x3x4=15^3 = 3375 we can expand this number to get some random x1-4. Though the equation is valid but you have to also satisfy the first requirement. Sorry, coudln't help u. Hope this might give some idea.
 

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