I was wondering if this is another valid way to write the proof: Prove the transitive property: If a|b and b|c, then a|c for all positive integers a,b,c. Let’s analyze our first given. We know that a|b. What does it mean for a to divide b? Let l be any positive integer. lℤ+. This means that a times some integer l equals b. Algebraically, we can say: al=b Manipulating this expression to solve for a, we find that: a=b/l Let’s analyze our second given: We know that b|c. What does it mean for b to divide c? Let m be any positive integer. mℤ+. This means that b times some integer m equals c. Algebraically, we can say: bm=c What are we trying to prove? We are trying to prove that a|c. What does it mean for a to divide c? Let n be any positive integer. nℤ+. This means that a times some integer n equals c. Algebraically, we can say: an=c We are trying to prove that a|c. We are trying to prove that an=c where n is some integer. This means we are trying to demonstrate that n=c/a is an integer. We have n=c/a. Recall that c=bm and a=b/l. Let’s replace c with bm. Let’s replace a with (b/l). n=(bm)/(b/l) We multiply bm by the reciprocal of (b/l). n=bm*(l/b) The bs here cancel, leaving us with: n=ml Recall that m is a positive integer. Recall that l is a positive integer. Positive integers are closed under multiplication. This means that the product of two positive integers yields a positive integer. Thus, n must be a positive integer. Since we have confirmed that n is a positive integer, we have confirm that a indeed divides c. Can you let me know whether or not this proof is valid? Thank you :)
So glad that you're back! I'm taking Calc I this semester and you've been a ton of help with my precalc and college algebra classes.
Any chance you could make some videos on optimization (calc I) or first/second derivative tests?
I was wondering if this is another valid way to write the proof:
Prove the transitive property:
If a|b and b|c, then a|c for all positive integers a,b,c.
Let’s analyze our first given.
We know that a|b.
What does it mean for a to divide b?
Let l be any positive integer.
lℤ+.
This means that a times some integer l equals b.
Algebraically, we can say:
al=b
Manipulating this expression to solve for a, we find that:
a=b/l
Let’s analyze our second given:
We know that b|c.
What does it mean for b to divide c?
Let m be any positive integer.
mℤ+.
This means that b times some integer m equals c.
Algebraically, we can say:
bm=c
What are we trying to prove?
We are trying to prove that a|c.
What does it mean for a to divide c?
Let n be any positive integer.
nℤ+.
This means that a times some integer n equals c.
Algebraically, we can say:
an=c
We are trying to prove that a|c.
We are trying to prove that an=c where n is some integer.
This means we are trying to demonstrate that n=c/a is an integer.
We have n=c/a.
Recall that c=bm and a=b/l.
Let’s replace c with bm.
Let’s replace a with (b/l).
n=(bm)/(b/l)
We multiply bm by the reciprocal of (b/l).
n=bm*(l/b)
The bs here cancel, leaving us with:
n=ml
Recall that m is a positive integer.
Recall that l is a positive integer.
Positive integers are closed under multiplication.
This means that the product of two positive integers yields a positive integer.
Thus, n must be a positive integer.
Since we have confirmed that n is a positive integer, we have confirm that a indeed divides c.
Can you let me know whether or not this proof is valid?
Thank you :)
legend 🦾