## Find largest x

For discussing Olympiad Level Number Theory problems
Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm

### Find largest x

Well.I am posting the first problem here,since first problem ofcourse easy.
Find the largest \$x\$ such that \$x+25\$ divides \$(x+2)^2\$
One one thing is neutral in the universe, that is \$0\$.

Zzzz
Posts: 172
Joined: Tue Dec 07, 2010 6:28 am
Location: 22° 48' 0" N / 89° 33' 0" E

### Re: Find largest x

here is the first reply of the first problem not sure... may be i have done some mistakes due to over excitement !

\[x+25|(x+2)^2\]
\[\Rightarrow x+25|x^2 + 4x + 4\]
\[\Rightarrow x+25|x^2 + 25x - 21x +4\]
\[\Rightarrow x+25|x(x+25) - 21x +4\]
\[\Rightarrow x+25| -21x +4\]
\[\Rightarrow x+25| 21(x+25) - 21x +4\]
\[\Rightarrow x+25| 529 \]

So, the highest possible value of \$x+25\$ is \$529\$

So, the largest value of \$x\$ is \$504\$
Every logical solution to a problem has its own beauty.

Moon
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
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### Re: Find largest x

Good Job!
Zzzz, I guess, I (we all) know you.
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin