prime number proof help

prove tha there is no prime number p such that is divisible by 54.


suppose

than

for some K in the naturals.


how would I proceed from here? very confused, I am suppose to come up with some sort of contradiction.

any help appreciated.

Look at it the "other way around":

Quote:

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Originally Posted by HallsofIvy

Look at it the "other way around":

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Not sure what you mean....

so

hence 9 divides p^2 ?

so 9 divides p ?

if p=9(mod54), p²=81(mod54)

p-9=kx54
p=9(1+6k)
p can't be prime.

Quote:

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Originally Posted by Tweety

prove tha there is no prime number p such that is divisible by 54.


suppose

than

for some K in the naturals.


how would I proceed from here? very confused, I am suppose to come up with some sort of contradiction.

any help appreciated.

---

p²=81+54k
p²=9(9+6k)
p=3(9+6k)^1/2

so p can't be prime even if (9+6k)^1/2 is an integer.

Quote:

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Originally Posted by Tweety

Not sure what you mean....

so

hence 9 divides p^2 ?

so 9 divides p ?

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No, either 9 divides p (if 9 also divides 6-k) or 3 divides p. Either way, p is not prime.