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# 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
3 years, 10 months ago

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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 ? · 3 years, 10 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? · 3 years, 10 months ago

How did you guys split 819529? · 3 years, 10 months ago

301 · 3 years, 10 months ago

My ans is also $$301$$ vaiia · 3 years, 10 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) . · 3 years, 10 months ago

I think it is not correct · 3 years, 10 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. · 3 years, 10 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. · 3 years, 10 months ago

$$0\leq b\leq 300$$ so b has $$301$$ choices. Jhon G. has already said that. · 3 years, 10 months ago

Granted, $$a$$ can be $$0$$, but when you type about non-negative, positive integers, one should assume that $$b \gt 0$$. · 3 years, 10 months ago

I request you to answer Yash T. & Bobby J. questions? It is also my question. · 3 years, 10 months ago

Also $$1+\sqrt{(-299)^2}$$, $$2+\sqrt{(-298)^2}$$, ... , $$299+\sqrt{(-1)^2}$$ for a total of $$598$$ ways :) · 3 years, 10 months ago

In my question there is no square.there is only $$\sqrt{a}$$ · 3 years, 10 months ago