Expression odd integer embodiment (not coprime to the modulus of linear congruence equation

# The meaning of problems

Given 2n integers, A _{. 1} , A ₂ , ......, A _n-, m _{. 1} , m ₂ , ......, m _n- ,

Seeking a smallest positive integer x, satisfies ≦ i ≦ n ∀, X m ≡ _I (A MOD _I )

If there is no solution "-1", or the output of the smallest positive integer x

data range:

1 ≤ a _i  ≤ 2 ³¹ -1

1 ≤ m _i <a _i

1 ≤ n ≤ 25

# Explanations

First consider the first two formulas:

x ≡  m₁ (mod a₁) ↔ x = k₁ * a₁ + m₁

x ≡  m₂ (mod a₂) ↔ x = k₂ * a₂ + m₂

k₁ * a₁ + m₁ =  k₂ * a₂ + m₂

k₁ * a₁ -k₂ * a₂ = m₂-m1

There is no solution that is determine what gcd (A ₁ , A ₂ ) | (m ₂ -m1)  is established

European expansion determined  K _{. 1}  * A _{. 1}  -k ₂  * A ₂  = GCD (A _{. 1} , A ₂ ) Solution K _{. 1} '' and K ₂ ''

k₁'=k₁'' * (m2-m1)/gcd(a₁,a₂) 

k₂'=k₂'' * (m2-m1)/gcd(a₁,a₂) 

K = a x _{. 1}  * A _{. 1}  + m _{. 1} or K = x ₂  * A ₂  + m ₂ can be obtained through the solutions of x

Wherein K _{. 1} the general solution K = _{. 1}  + K * (A ₂ / GCD (A _{. 1} , A ₂ )) clearly K _{. 1} when the acquired minimum x minimum

But when the formula behind and merged with the new constraints,

The minimum value of x may change it to not find a specific x,

Obtained by k ₁ or k ₂ constituting a new congruence equation general solution of

The K _{. 1} the general solution K = _{. 1} '  + K * (A ₂ / GCD (A _{. 1} , A ₂ )) into X = K _{. 1}  * A _{. 1}  + m _{. 1}

x = ( k₁' + k* ( a₂/gcd(a₁,a₂) ) ) * a₁ + m₁

x = k * ( a₁ * a₂ )/gcd(a₁,a₂) + k₁'*a₁+m₁

a'=( a₁ * a₂ )/gcd(a₁,a₂) , m'=k₁'*a₁+m₁

x = k * a' + m'

K = X ₃  * of a ₃  + m No ₃

Continuously merging into k-1, determined in accordance with the smallest x k is the general solution,

k = k₀ + t * (a_n/gcd(a',a_n)) 

Clearly K ₀ % (A _n- / GCD (A ', A _n- )) i.e., the smallest solution