1 Prove or Disprove a If abc are integer i Prove or disprove

1. Prove or Disprove

(a) If a,b,c are integer

   i. Prove or disprove : If gcd(a,b)=gcd(b,c)=5 then gcd(a,c)=5

   ii. If a|b and b|c then a|(c-2b)

(b) For all integers n, if n^2 is divisible by 5, then n is divisible by 5

Solution

a(I)

If g(b,c)=5 then b= 5y, c=5z

and if g(a,b) then a=5x, b=5y

So all the three values of a,b,c are multiple of 5.
Now we can not say from the above conclusion that a,c will also have greatest common divisor as 5.
for eg, a=10,b=5,c=20
gcd(a,b)=5 and gcd(b,c)=5
but gcd(a,c)=10.

So diapprove with the statement : gcd(a,b)=gcd(b,c)=5 then gcd(a,c)

(b) if n^2 is divisible by 5 then it states that (n*n) is divisible by 5.
that is n has a factor of 5. so if n^2 is divisible by 5 then n is also divisible by 5.
eg : 15^2 =15*15 =225, is divisible by 5 and so is 15 divisible by5 .
It is so because when we prime factorize 15 we get 3*5 and for n^2 we get 3^2 * 5^2. so as there is power 2 for 5 it is always divisible by 5.