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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 ? Sadman Sakib · 3 years, 10 months ago

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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? Bobby Jim · 3 years, 10 months ago

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

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301 Adeeb Zaman · 3 years, 10 months ago

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@Adeeb Zaman My ans is also \(301\) vaiia Mashrur Fazla · 3 years, 10 months ago

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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) . Indulal Gopal · 3 years, 10 months ago

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@Indulal Gopal I think it is not correct Mashrur Fazla · 3 years, 10 months ago

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@Mashrur Fazla Is the answer Sum of total number of factors=\(300\) as b is a non-negative positive integer. so there are 301 possibilities. John Gray · 3 years, 10 months ago

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@Mashrur Fazla How Mashrur you got 301? Explain. While you post a problem dont comment like " I THINK IT IS 301". Be more transparent in reply. Indulal Gopal · 3 years, 10 months ago

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@Indulal Gopal \(0\leq b\leq 300\) so b has \(301\) choices. Jhon G. has already said that. Mashrur Fazla · 3 years, 10 months ago

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@Mashrur Fazla Granted, \(a\) can be \(0\), but when you type about non-negative, positive integers, one should assume that \(b \gt 0\). Ton De Moree · 3 years, 10 months ago

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

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

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@Ton De Moree In my question there is no square.there is only \(\sqrt{a}\) Mashrur Fazla · 3 years, 10 months ago

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@Mashrur Fazla Ah, I see, my mistake :) Ton De Moree · 3 years, 10 months ago

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