Make It One CodeChef Solution

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Make It One CodeChef Solution

Make It One CodeChef Solution

• Problem

  Chef is given a positive integer $N$ and we have to convert it to $1$ using some operations.\newline

  In one operation, Chef can do the following:\newline

  • Choose a prime number $p$ such that $p$ $|$ $N$ $(p$ divides $N)$.
  • Replace $N$ with \frac{N}{p}pN​.

  However, Chef is also given some triggers of the form $(a,b)$ ($a$ is prime). Thus, if we divide by $p$ in some operation, we also have to multiply by $x$ for all triggers of the form $(p,x)$.

  For example: Consider $N=18$, and the triggers $(3,4)$ and $(3,2)$ exist. In one operation, if we choose to divide by $3$, we have to multiply the result with $4$ and $2$. In other words, (18 \mapsto \frac{18}{3} \cdot 4 \cdot 2)=48(18↦318​⋅4⋅2)=48. Thus, $18$ is converted to $48$.

  Given a number $N$ and $M$ triggers of the form $(a,b)$ (where $a$ is always prime), find the minimum number of operations required to convert $N$ to $1$ modulo $998244353$. Print $-1$ if it is not possible to do so.

  Input Format

  • The first line of each input contains two integers $N$ and $M$ – the initial integer and the number of triggers given to the Chef.
  • The next $M$ lines describe the triggers. The i^{th}ith of these $M$ lines contains two space-separated integers $a$ and $b$, denoting that Chef has to multiply by $b$ if he choses to divide by $a$ in some operation.
  • It is guaranteed that (a_i, b_i) \ne (a_j, b_j)(ai​,bi​)=(aj​,bj​) for i \ne ji=j.

  Output Format

  Output on a new line, the minimum number of operations required to convert $N$ to $1$, or print $-1$ if it is not possible to do so. In case it is possible to convert $N$ to $1$, the minimum number of operations can be large, so output it modulo $998244353$.

  Constraints

  • 1 \leq N \leq 10^61≤N≤106
  • 0 \leq M \leq 10^50≤M≤105
  • 1 \leq a \leq 10^61≤a≤106
  • 1 \leq b \leq 10^61≤b≤106
  • $a$ is a prime number.

  Sample 1:

  Input

  Output

  6 1
  2 3
  

  3
  

  Explanation:

  We can convert $6$ to $1$ using $3$ operations in the following way:

  • Divide $6$ by $3$. Since no trigger of the form $(3,x)$ exists, $6$ converts to \frac{6}{3}= 236​=2.
  • Divide $2$ by $2$. Since $(2,3)$ exists, $2$ is converted to 2 \mapsto \frac{2}{2} \cdot 3 = 32↦22​⋅3=3. – Divide $3$ by $3$. Since no trigger of the form $(3,x)$ exists, $3$ converts to \frac{3}{3}= 133​=1.

  It can be proven that we cannot convert $6$ to $1$ in less than $3$ operations.

  Sample 2:

  Input

  Output

  4 3
  2 5
  2 3
  3 7
  

  8
  

  Sample 3:

  Input

  Output

  2 3
  2 3
  3 5
  5 3
  

  -1
  

  Explanation:

  We can not convert $2$ to $1$ using any finite number of operations

 

SOLUTION

Here Discuss About Make It One CodeChef Solution

SOLUTION

Make It One CodeChef Solution

SOLUTION

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