Good numbers Problem | PRMO-2019 | Problem 12

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Integer | PRMO | Problem-12
- Key Concepts
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Try this beautiful problem from PRMO, 2019 based on Good numbers.

Good numbers Problem | PRMO | Problem-12

A natural number $k > $ is called good if there exist natural numbers
$a_1 < a_2 < ………. < a_k$

$\frac{1}{\sqrt a_1} +\frac{1}{\sqrt a_2}+................... +\frac{1}{\sqrt a_k}=1$

Let $f(n)$ be the sum of the first $n$ good numbers, $n \geq 1$. Find the sum of all values of $n$ for which
$f(n + 5)/f(n)$ is an integer.

$20$
$18$
$13$

Key Concepts

Number theory

Good number

Integer

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Answer:$18$

Source

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First hint

A number n is called a good number if It is a square free number.

Let $a_1 ={A_1}^2$,$a_2={A_2}^2$,..................$a_k={A_k}^2$
we have to check if it is possible for distinct natural number $A_1, A_2………….A_k$ to satisfy,
$\frac{1}{A_1}+\frac{1}{A_2}+...........+\frac{1}{A_k}=1$

Can you now finish the problem ..........

Second Hint

For $k = 2$; it is obvious that there do not exist distinct$ A_1, A_2$, such that $\frac{1}{A_1}+\frac{1}{A_2}=1 \Rightarrow 2$ is not a good number

For $k = 3$; we have $\frac{1}{2} +\frac{1}{3}+\frac{1}{6}=1 \Rightarrow 3$ is a good number.

$\frac{1}{2}+\frac{1}{2}\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$ $\Rightarrow 4$ is a good number

Let $k$ wil be a good numbers for all $k \geq 3$

$f(n) = 3 + 4 +… n$ terms =$\frac{n(n + 5)}{2}$
$f(n + 5) =\frac{(n + 5)(n +10)}{2}$

$\frac{f(n+5}{f(n)}=\frac{n+10}{n}=1+\frac{10}{n}$

Can you finish the problem........

Final Step

Therefore the integer for n = $1$, $2$, $5$ and $10$. so sum=$1 + 2 + 5 + 10 = 18$.