# Combinatorics in Number of Factor's Sum

The sum of the total number of factors of $999000$, $816480$ and $819529$is n. How many ways can n be written as $\sqrt{a}+b$ where b is a non-negative integer?

Note by Mashrur Fazla
6 years, 8 months ago

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How did you guys split 819529?

- 6 years, 8 months ago

999000=2^3 X 3^3 X 5^3 X 37

816480=2^5 X 3^6 X 5 X 7

But how did you find out 819529=743 X 1103?

- 6 years, 8 months ago

If b is a non-negative integer then $b \geq 0$ and it gives the answer 301. Again,if b is a positive integer then $b > 0$ and we can find 300 ways, Why non-negative positive ?

- 6 years, 8 months ago

301

- 6 years, 8 months ago

My ans is also $301$ vaiia

- 6 years, 8 months ago

Total number of ways = 299

Solution:

999000=2^3 X 3^3 X 5^3 X 37 ; total number of factors = 128

816480= 2^5 X 3^6 X 5 X 7 ; total number of factors = 168

819529 = 743 X 1103 ; total number of factors = 4

Sum of total number of factors = n= 300 ; It can be written as 1+ sqrt(299^2),

2+sqrt(298^2),............,299+sqrt(1^2) .

- 6 years, 8 months ago

I think it is not correct

- 6 years, 8 months ago

Is the answer Sum of total number of factors=$300$ as b is a non-negative positive integer. so there are 301 possibilities.

- 6 years, 8 months ago

How Mashrur you got 301? Explain. While you post a problem dont comment like " I THINK IT IS 301". Be more transparent in reply.

- 6 years, 8 months ago

$0\leq b\leq 300$ so b has $301$ choices. Jhon G. has already said that.

- 6 years, 8 months ago

Granted, $a$ can be $0$, but when you type about non-negative, positive integers, one should assume that $b \gt 0$.

- 6 years, 8 months ago

I request you to answer Yash T. & Bobby J. questions? It is also my question.

- 6 years, 8 months ago

Also $1+\sqrt{(-299)^2}$, $2+\sqrt{(-298)^2}$, ... , $299+\sqrt{(-1)^2}$ for a total of $598$ ways :)

- 6 years, 8 months ago

In my question there is no square.there is only $\sqrt{a}$

- 6 years, 8 months ago

Ah, I see, my mistake :)

- 6 years, 8 months ago