$p_1+p_2+p_3 = 1234$ and $p_1p_2p_3 = 87654321$

Question

Is there any $3$ primes such that their sum is $1234$ and product is $87654321$.

I can factorize $87654321$ by wolfram-alpha - $3^2 \times 1997 \times 4877$. So there is no solution.
But I dont know how I can do this in hand. Is there any simpler way to prove this?

Answer 1

The sum of three odd numbers is odd.

$1234$ is even, so one of the three primes has to be $2$.

However, this is impossible because $87654321$ is odd.

Answer 2

Eliminate one variable from both equations, form a quadratic equation and see its discriminant. It will be clear that there is no solution.

Answer 3

No. The sum of three primes is even so one of them is $2$, but in this case the product of them will be even.

Answer 4

$87654321$ is divisible by $9$, so two of the three primes would have to be $3$. But then the third one would have to be $1234-3-3=1228$ which is not a prime, thus the problem has no solutions.