I need help proving the following problems Prove the followi

I need help proving the following problems.

Prove the following p abc then pla or pla or pc +or positive a,b f alo and bla then a=b nwest terms are uni 71. e: it (a,b) 1, (c,d] and b=d na- edi 1 and art cid then . c 1 and a/d c/d then a

Solution

(1)

we are given that

if p|abc

it means that p divides abc

we know that

abc=a*b*c

so, \'p\' will divide either one of them

if p divides \'a\' ---------> p|a

if p divides \'b\'--------->p|b

if p divides \'c\'---------->p|c

Hence , if p|abc , then p|a or p|b or p|c.........Answer

(2)

if (d,a)=1 .it means that there is no common factor between d and a except 1

and d|ab ...it means that d divides ab

ab=a*b

since, d can not divide \'a\'

so, \'d\' will divide \'b\'

Hence , d|b..........Answer

(3)

if a|b .......It means that b=k*a

if b|a...it means that a=m*b

by using two equatons

we get

b=k*m*a

k*m=1

so, b=a

(4)

(a,b)=1 .it means that no term are factor brtween a and b

(c,d)=1 ....it means that no terms are factor between c and d

a/b = c/d

a/b=1

a=b

c/d=1

c=d