1. ## any help plz

hi every one
can any buby help me with these questions?

1. If a,b are in N*, if gratest common div.(a,b) & smallest com. mult.(a,b) are squares, show that a,b are squares.

2. Show that for every m>0 , a?1 we have

( a^m -1 / a-1 , a-1 ) = ( a-1 , m ).

3. show that any two terms in the sequance: 2+1, 2^2+1, 2^4+1, .. , 2^2n+1 are coprime .

2. Originally Posted by miss blue
hi every one
can any buby help me with these questions?

1. If a,b are in N*, if gratest common div.(a,b) & smallest com. mult.(a,b) are squares, show that a,b are squares.

2. Show that for every m>0 , a?1 we have

( a^m -1 / a-1 , a-1 ) = ( a-1 , m ).

3. show that any two terms in the sequance: 2+1, 2^2+1, 2^4+1, .. , 2^2n+1 are coprime .

let

$\displaystyle a=p_1\times p_2 \times ... \times p_n$

$\displaystyle b=q_1\times q_2 \times ...\times q_m$

such that p,q are prime numbers
g.c.d(a,b) the product of the similar primes in a and b say

$\displaystyle g.c.d(a,b)=p_1\times p_2 \times ... \times p_i=q_1\times q_2 \times ... \times q_i=x^2$

x is integer number, there exist such x since g.c.d is square

$\displaystyle l.c.m(a,b)=g.c.d(a,b)\times (p_{i+1}\times ...\times p_n)(q_{i+1}\times ...\times q_m)=y^2$

y is integer, there exist such y since l.c.m is square

note that
$\displaystyle \left(\frac{a}{g.c.d(a,b)},\frac{b}{g.c.d(a,b)}\ri ght)=1$ in other word

$\displaystyle (p_{i+1}\times ...\times p_n)$ and $\displaystyle (q_{i+1}\times...\times q_m)$ are relatively primes

$\displaystyle g.c.d(a,b)\times (p_{i+1}\times ...\times p_n)(q_{i+1}\times ...\times q_m)=y^2$

g.c.d(a,b) , $\displaystyle (p_{i+1}\times ...\times p_n)$ ,$\displaystyle (q_{i+1}\times ...\times q_m)$ all of them are squares

say
$\displaystyle (p_{i+1}\times ...\times p_n)=t^2$

$\displaystyle q_{i+1}\times ...\times q_m)=s^2$

s,t are integers now

$\displaystyle a=x^2(t^2)$

$\displaystyle b=x^2(s^2)$

so a,b are squares

3. thx alot Amer

but plz if u have a selution for the athors be generous
ant kareem wa n7n nastahal

thx

4. Originally Posted by miss blue
hi every one
can any buby help me with these questions?

1. If a,b are in N*, if gratest common div.(a,b) & smallest com. mult.(a,b) are squares, show that a,b are squares.

2. Show that for every m>0 , a?1 we have

( a^m -1 / a-1 , a-1 ) = ( a-1 , m ).

3. show that any two terms in the sequance: 2+1, 2^2+1, 2^4+1, .. , 2^2n+1 are coprime .
in the second question you want to prove that

$\displaystyle \left( \frac{a^{m-1}}{a-1} , a-1 \right) = (a-1 , m )$

or

$\displaystyle \left( \frac{a^m -1 }{a-1} , a-1 \right) = ( a-1 , m )$

m>0 , is there any conditions on a

5. yessss sorry
i copied it wrong

a>1

thx

6. $\displaystyle \left(\frac{a^m-1}{a-1} , a-1 \right) = (a-1 , m )$

note that

$\displaystyle (a,b)=(r,b)$

r is the reminder such that $\displaystyle a=xb+r$

now

$\displaystyle \frac{a^m-1}{a-1} = a^{m-1} + a^{m-2} + ...+ 1$

but

$\displaystyle ( a^{m-1} + + ...+ 1 , a-1 ) = (a-1 , r )$

r is the reminder

$\displaystyle a^{m-1} + + ...+ 1 = (a-1)(s) + r$

to find r sub a=1 you will have

$\displaystyle 1+1+...+1 = 0(s) + r \Rightarrow r=m$

the proof is finished