100 followers problem!

Number Theory Level 3

The diagonal of the square with side 100 cm can be expressed as $a\times\sqrt b cm$ and b is Square Free.

Then X=ab

Let Y be the number of trailing zeroes in 100!

Let the roots of the quadratic equation ${x}^{2} + 100 x + 819$ be p and q

Then ${p}^{2} + {q}^{2} = Z$

Let A be the number of trailing zeroes in (X+Y+Z)!

Find $X+Y+Z+A$

The answer is 10729.

3 solutions

Mehul Arora
Mar 17, 2015

I will just give away the answers.... I encourage you to solve them on your own.

The diagonal of the square can be expressed as $100\times\sqrt 2$ . Hence, X=200
The number of trailing zeroes in 100! is 24
The roots are -91 and -9... Therefore, Z= 8362
X+Y+Z = 8586. Therefore, the number of trailing zeroes in 8586! is 2143. Hence, A= 2143.

Adding them all up, you get 10729.

Cheers! :)

A faster way to calculate Z : ( $p+q)^2 -2pq=100^2 - 2*819$

Tarun Singh - 6 years, 2 months ago

That's what is newton's identity!

Nihar Mahajan - 6 years, 2 months ago

Sorry I was confused !

A Former Brilliant Member - 6 years, 2 months ago

@A Former Brilliant Member – Well , refer this page . I hope its clear now . Thanks!

Nihar Mahajan - 6 years, 2 months ago

Thanks for not changing the answer , else I would have had to go through all the calculations again!

A Former Brilliant Member - 6 years, 2 months ago

No worries.The question is still the original and the same! No need to thank me for that!

Mehul Arora - 6 years, 2 months ago

Yeah , ok :)

A Former Brilliant Member - 6 years, 2 months ago

Got it right! Same method , the only difference is in 3rd step I used newton's identity...

Nihar Mahajan - 6 years, 2 months ago

For trailing zeroes, I used Legendré formula for the greatest power of 5 in $n!$ to find number of trailing zeroes in both cases, Vietà's formula for the algebra problem and Pythagoras theorem for the geometry problem.

Venkata Karthik Bandaru - 6 years, 2 months ago

Ramiel To-ong
May 12, 2016

nice analysis. same as mine

Mrigank Krishan
Mar 21, 2015

There is an error in the first para,

Diagonal can be also written as 50 root 8

Yes sir... But the simplest answer is 100 root 2 as a diagonal is side of the square multiplied root 2

Mehul Arora - 6 years, 2 months ago

I think u must mention in the question that b is 'square free' or 'independent of square" .

Nihar Mahajan - 6 years, 2 months ago

100 followers problem!

The answer is 10729.

3 solutions

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