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# Good numbers Problem | PRMO-2019 | Problem 12

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

Answer:$$18$$

PRMO-2019, Problem 12

Pre College Mathematics

## Try with Hints

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 ..........

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........

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

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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

Answer:$$18$$

PRMO-2019, Problem 12

Pre College Mathematics

## Try with Hints

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 ..........

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........

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

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