# Math Help - Number theory (primes)

1. ## Number theory (primes)

*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

2. Originally Posted by blazingmath
*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
...
If GCD(a, $p^2$) = p then a = p or a = mp (a multiple of p, m<p )

If GCD(b, $p^3$) = $p^2$ then b = $p^2$ or b = k $p^2$ (a multiple of $p^2$, and $k \neq p$)

ab = $mp \cdot kp^2 = kmp^3$
Thus,
GCD( $kmp^3,p^4) \, = \, p^3$

$a+b = mp + kp^2 \, = \, p(m+kp)$
&
GCD $( p(m+kp), p^4) \, = \, p$

.

3. ## primes

thank u very much for ur reply...really thanks..god bless you