Assume that a,b,n areintegers, that a and b are coprime, that adivides n and b d

Question

Assume that a,b,n areintegers, that a and b are coprime, that adivides n and b divides n.

Show that ab divides n

Explination:

a divides n means that n=ak for some intger k
ab divides n means that is, n=abl for some integerl
a and b are coprime means that that greatest common divisorof a and b is 1

...I thank you in advance for any help at all! Assume that a,b,n areintegers, that a and b are coprime, that adivides n and b divides n.

Show that ab divides n

Explination:

a divides n means that n=ak for some intger k
ab divides n means that is, n=abl for some integerl
a and b are coprime means that that greatest common divisorof a and b is 1

...I thank you in advance for any help at all!

Explanation / Answer

QuestionDetails: Assume that a,b,n areintegers, that a and b are coprime, that adivides n and b divides n.

Show that ab divides n
A|N.....N=P*A..................................................1
WHERE P IS AN INTEGER
B|N......N=Q*B...............................................2
WHERE Q IS AN INTEGER
A AND B ARE PRIME TO EACH OTHER....(A,B)=1...OR
WE CAN FIND 2 INTEGERS X AND Y SUCH THAT
AX+BY=1..............................................3
TST AB|N
THAT IS TST
N=R*AB ..............................................4
WHERE R IS AN INTEGER
MULTIPLYING EQN.3 WITH N AN INTEGER WE GET
AXN+BYN=N
SUBSTITUTING FROM 1 AND 2 FOR N
AXQB+BYPA=N
AB(QX+PY)=N
N=AB*R WHERE R=PY+QX IS AN INTEGER
HENCE AB|N .... PROVED

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