Question

Prove or Disprove: You are given the group of integers (a1,..., an). If t/ai
for every 0 <= i <= n then t/gcd(a1, a2,..., an).

Answer 1

\[\Large t \Bigg| a_{i}\]
means
\[\Large t*k_{i} = a_{i}\]
where the set of k values \(\Large (k_1,k_2,k_3,...,k_n)\) are all integers


Basically this means that t is a common factor of all the terms (a1, ..., an). It is not guaranteed to be the greatest common factor of the set (a1,...,an). The GCD could be equal to t*m where m is some positive number. If m = 1, then t is the GCD.


So this is an informal way to prove that if t divides (a1,...,an) then t also divides the GCD of that set of numbers.


In other words, if t is a factor of (a1,...,an) then t is a factor of the GCD of that set of numbers.

Answer 2

if \(t|a_i\) then it follows \(a_i=tb_i\) where \(b_i\) is an integer. it follows that \(\gcd(a_1,\dots,a_n)=\gcd(tb_1,\dots,tb_n)=t\cdot\gcd(b_1,\dots,b_n)\) so \(t|\gcd(a_1,\dots,a_n)\)

Answer 3

thanks