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RD19 - Editorial

PROBLEM LINK:

Practice

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Author: nileshjha19

Tester: Suchan Park

Editorialist: Suchan Park

DIFFICULTY:

Very Easy

PREREQUISITES:

How to compute GCD (Euclidean Algorithm)

PROBLEM:

You are given a sequence of integers of length $N$. You are allowed to delete up to $N-1$ elements in this sequence. Your goal is to make the GCD of the resulting sequence 1. Find whether it is possible, and if it's possible, find the minimum number of deletions.

QUICK EXPLANATION:

It makes no sense to remove any elements, since any such operation doesn't decrease the GCD and increase the number of deletions. Therefore the answer is $0$ if the GCD of the original array is 1, $-1$ otherwise.

EXPLANATION:

From the definition of GCD, we know that

$$\texttt{gcd}(a, b) \le \texttt{min}(a,b)$$

This means whenever we apply the GCD operation, the value does not increase. Therefore, the longer the sequence is, the smaller the GCD becomes.

Since we want to minimize the GCD and the number of deletions, we can see the best strategy is not deleting any of the elements!

If the GCD of the given sequence is 1, we are done. Otherwise, the GCD will not be 1 even if we delete some elements (since the GCD will not decrease), it is impossible to make the GCD 1.

Therefore the answer is $0$ if the GCD of the original array is 1, $-1$ otherwise.

We can compute the GCD of the whole array by:

$$\texttt{gcd}(A_1,\texttt{gcd}(A_2,\texttt{gcd}(A_3,\cdots,\texttt{gcd}(A_{n-1}, A_{n})\cdots))$$

AUTHOR'S AND TESTER'S SOLUTIONS:

Author's solution can be found here

Tester's solution can be found here

Editorialist's solution can be found here.

This question is marked "community wiki".

asked 19 May, 06:45

tncks0121's gravatar image

2★tncks0121
137
accept rate: 0%

edited 22 May, 18:20

admin's gravatar image

0★admin ♦♦
19.3k348495534


Is there a comparable (more difficult) problem asking for the maximum number of elements that can removed and leave the gcd at $1$, or perhaps at whatever $\gcd(\{A_i\})$ is?

link

answered 14 Sep, 22:32

joffan's gravatar image

5★joffan
4186
accept rate: 9%

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question asked: 19 May, 06:45

question was seen: 1,166 times

last updated: 14 Sep, 19:41