GATE CSE 2016 Set 1 | Question: 28

Question

A function $f: \Bbb{N^+} \rightarrow \Bbb{N^+}$ , defined on the set of positive integers $\Bbb{N^+}$, satisfies the following properties:

                    $f(n)=f(n/2)$   if $n$ is even

                    $f(n)=f(n+5)$  if $n$ is odd

Let $R=\{ i \mid \exists{j} : f(j)=i \}$ be the set of distinct values that $f$ takes. The maximum possible size of $R$ is ___________.

Comments

....….………..................…......…....………….…….…………..

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Such Questions make the preparation journey more interesting :D

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Haywire

😂

Answer 1

Let us assume: $f(1) = x.$

Then, $f(2) = f(2/2) = f(1) = x$
$ f(3) = f(3+5) = f(8) = f(8/2) = f(4/2) = f(2/1) = f(1) = x.$

Similarly, $f(4) = x$
$f(5) = f(5+5) = f(10/2) = f(5) = y.$

So, it will have two values. All multiples of $5$ will have value $y$ and others will have value $x.$

Comments

very nice explanation.

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Whoever is calling it explanation they are wrong. There should be a proof that this should be true always. How you know that it will not faile for say 47 or 4775 in thoery you should have known it before hand otherwise I don’t think you can solve this one

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Got it!🙂🙂

Answer 2

Answer is 2
Its Saying we have 2 domains
N⁺ → N⁺

1. So F(1) = F(6) = F(3) = F(8) = F(4) = F(2) = F(1)....It Repeats...  Now F(7) = F(12) = F(6)...Again repeats both above are same...Since F(6) matches in both so same both belongs to same value.We are not getting F(5) above
2. Now F(5) = F(10) = F(5)..Repeats ...We can see we have different value for multiples of 5 and other natural numbers.

Comments

Thank you for so nice explanation.

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@ Deepesh Kataria .where is f(9)

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f(9) = f(9+5)= f(14)= f(14/2)= f(7)

now f(7) = f(7+5)= f(12)=f(12/2)= f(6)=f(3)=f(8) etc.

Answer 3

Answer 2..

for multiples of 5.. f(5)=f(10)...
and one for rest of the numbers in N.

Comments

@akriti, see defination of function again, it's in recursive form f(n) = f(n/2) ( see the recursion)  f(1) = f(6) so in order to cal. f(1) we need to agin cal f(6) now this will give f(3) now again we need to fine f(3) which will be f(8).now f(8)will give f(4) and then f(4) will give f(2) and then f(2) -> f(1) so in this way it repeats itself. see the function again, the catchy thing is the recursion part.

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ooh..i missed the recursion part..thanks @mohit..:-)

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Thanks for this.…

I was also stuck here

Such silly things 🤦

Answer 4

http://math.stackexchange.com/questions/2118739/finding-recursive-function-range/2118749

We will use strong Induction Hypothesis to proof this.

Suppose that $f(1) = a$ and $f(5) = b$. It is clear that $$f(5n) = b$$ for all $n$. We'll prove by induction that for all $n \ne 5k$, $f(n) = a$.
First note that
$$f(2) = f(\frac{2}{2}) = f(1) = a,$$
$$f(3) = f(3+5) = f(8) = f(4) = f(2) = a,$$
$$f(4) = f(2) = a.$$
Now suppose $n = 5k + r$, where $0 \lt r \lt 5$, and for all $k\lt n$, $n$ is not divisible by $5$ bcoz $r \neq 0$
Note that if  $n$ is not divisible by $5$ then $n-5$ is also not divisible by $5$. Because $n-5 = 5(k-1) + r$, again $r \neq 0$.
And also Note that $\frac{n}{2}$ is not divisible by $5$, bcoz if it were divisible by $5$, this will make $n$ divisible by $5$. 

Base case:  $f(1)=f(2)=f(3)=f(4)=a$ [already solved for base cases above]
Incuctive step: Now suppose $n = 5k + r$, where $0 \lt r \lt 5$, and for all $m\lt n$ which are not divisible by $5$, $f(m) = a$.
($m$ already covers $n-5$ and $\frac{n}{2}$)
If $n$ is odd, $f(n) = f(n-5)$, and by induction hypothesis, $f(n-5) = a$, so we get $$f(n) = a.$$
If $n$ is even,  $f(n) = f(n/2)$, and by induction hypothesis, $f(n/2) = a$, so we get $$f(n) = a.$$

Comments

Best solution by using mathematical Induction.Thanks :-)

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Hi Sachin,

How "If n is odd, f(n)=f(n−5)" this is true?

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@Sourav Basu
Yes.

$\because f(n)=f(n+5)$.

Putting $n=n-5$ [where $n>5$] will yield $f(n-5)=f(n-5+5)=f(n)$