## BdMO National Junior 2020 P11

Mursalin
Posts: 67
Joined: Thu Aug 22, 2013 9:11 pm

### BdMO National Junior 2020 P11

$$n$$-এর সকল সম্ভাব্য মানের যোগফল বের করো যাতে $$n$$, $$n^2+10$$, $$n^2-2$$, $$n^2-8$$, $$n^3+6$$-এর সবগুলো মৌলিক সংখ্যা হয়। (হিন্ট: এরকম অন্তত একটি $$n$$ রয়েছে)।

Find the sum of all possible $$n$$ such that $$n$$, $$n^2+10$$, $$n^2-2$$, $$n^2-8,$$ $$n^3+6$$ are all prime numbers. (Hint: there is at least one such $$n$$.)
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Enthurelxyz
Posts: 17
Joined: Sat Dec 05, 2020 10:45 pm
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### Re: BdMO National Junior 2020 P11

Let, $n\neq 7$

So, $n \equiv 1,2,3,4,5,6 (mod$ $7)$ n^2 \equiv 1,2,4 (mod7)$. If$n^2 \equiv 1 (mod7)$then$n ^2-8\equiv 0 (mod7)$. So,$n^2=115$but$15$is not a perfect square. If$n^2\equiv 2 (mod7)$then$n^2-2=7 n=3but $3^2-8=1$ which is not a prime.

If $n^2 \equiv 4 (mod$ $7)$ then $n^2+10\equiv 0(mod$ $7)$ $n^2+10=7$ but it has no real solution.

If $n=7$ then $n^2+10,n^2-2,n^2-8,n^3+6$ are all prime.

So, $n=7$ and answer is $7$.
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