Proof That if Relatively Prime Numbers Divide a Third Number, Then So Does Their Product
If
\[a | c, \; b | c \]
with the greatest common divisor of \[a, \; b\]
equal to 1, then \[ab | c\]
.Proof
\[a | c \rightarrow ar=c\]
for some integer \[r\]
.\[b | c \rightarrow bs=c\]
for some integer \[s\]
.The greatest common divisor of
\[a, \; b\]
is 1, so there exist integers \[m,n\]
such that \[am+bn=1\]
.Multiply through by
\[c\]
to get\[amc+bnc=c\]
Substitute for
\[c\]
from above on the left hand side.\[ambs+bnar =c \rightarrow ab(ms+nr)=c \rightarrow (ab) | c\]
