we let a,b,c belongs to N. if (a,b) = 1 and a | (bc) prove that a|c.
2006-09-26 17:57:41 · 4 answers · asked by David F 2 in Science & Mathematics ➔ Mathematics
Your forgot the D, D(a,b) is 1
albc => a divides b or a divides c or both. But a doesnt divide b, nor has any common factor with b, so a must divide c.
Or:
Lets say that a doesnt divide c, since a doesnt divide b, a cant divide bc
Ana
2006-09-26 18:02:42 · answer #1 · answered by MathTutor 6 · 0⤊ 2⤋
This is an exercise in factorization into primes. a|(bc) means ad=bc for some integer d. Factor ad into primes and factor bc into primes. The number a has no primes in common with b, since (a,b)=1, as this is the definition of the symbols. So, all the prime powers that appear in the factorization of b on the right-hand side of this equality also appear on the left-hand side of this equality and do not appear in the factorization of a. So if we cancel all the factors of b from both sides, we get ea=c for some *integer* e. This is exactly the statement that a|c.
2006-09-27 12:10:36 · answer #2 · answered by just another math guy 2 · 1⤊ 1⤋
If (a,b) =1 then a does not divide b.
Now if a | bc then a is either equal to c or a | c.
Very simple.
2006-09-27 09:38:47 · answer #3 · answered by Anonymous · 0⤊ 0⤋
Proof:
If (a,b) = 1
Then a/(bc)
=> 1/(1c) by substitution
=> 1/c by the existence of multiplicative identity, c=1c
=> a/c since a=1.
2006-09-27 01:11:25 · answer #4 · answered by kwatog 2 · 0⤊ 3⤋