*Evaluate (ab, p^4) and (a+b,p^4) given that (a,p^2)=p and (b, p^3)=p^2 where p is a prime
.................note: i just use ^...it's actually p above 4...i can't draw....thanks..i mean p to the power of 4
*Evaluate (ab, p^4) and (a+b,p^4) given that (a,p^2)=p and (b, p^3)=p^2 where p is a prime
.................note: i just use ^...it's actually p above 4...i can't draw....thanks..i mean p to the power of 4
If GCD(a,$\displaystyle p^2$) = p then a = p or a = mp (a multiple of p, m<p )
If GCD(b,$\displaystyle p^3$) = $\displaystyle p^2$ then b = $\displaystyle p^2$ or b = k$\displaystyle p^2$ (a multiple of $\displaystyle p^2$, and $\displaystyle k \neq p $)
ab = $\displaystyle mp \cdot kp^2 = kmp^3 $
Thus,
GCD($\displaystyle kmp^3,p^4) \, = \, p^3 $
$\displaystyle a+b = mp + kp^2 \, = \, p(m+kp) $
&
GCD$\displaystyle ( p(m+kp), p^4) \, = \, p $
.