- #1

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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

- Thread starter kurt.physics
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- #1

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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

- #2

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does anyone have an answer?

- #3

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anyone out there???

- #4

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(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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