Can you prove that ab/m is a common divisor of a and b?

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In summary, if a is a common multiple of b and m divides ab, then ab/m is a common divisor of a and b.
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Homework Statement


Let a,b and m be positive integers. If m is a common multiple of a and b and m divides ab, then ab/m is a common divisor of a and b.


Homework Equations


Definition: A common multiple of a and b is any integer divisible by both a and b.
Definition: A common divisor of a and b is any integer that divides both a and b.

The Attempt at a Solution


1. Let a,b,m be positive integers. suppose m is a common multiple of a and b and m divides ab.
2. By definition m does not equal 0, a divides m and b divides m so a and b both do not equal 0.
3. There exist integers S,D such that m=aS and m=bD
4. From (1) m divides ab so there exists an integer F such that ab=Fm
5. ab=Fm=aSF and bDF
6. ab/m=aSF/aS and bDF/bD, both Give F
7. a and b are both divisible by F so ab/m is a common divisor of a and b.

I believe I have an error between 6 and 7 because it seems as though I am jumping to conclusions about both being divisible by F.
Thank you
 
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  • #2
Step 7. Why are a and b divisible by F? From the previous step ab is divisible by F but, from this, you cannot say that a and b are also divisible by F.
 
  • #3
Therein lies my question of what should I do to correct step 7? I am not sure what to do. It is necessary to show that F divides both a and b right?
 
  • #4
But that is not possible in general. Try this instead: prove that ab divides ma and mb. What could you conclude from that?
 
  • #5
This just tells me that b divides m and that a divides m which is what I already know. I think I am reading what you asked incorrectly.
 
  • #6
If ab|ma, then ma = k(ab), which implies that a = k(ab/m) (why can we say that?); what does this mean? If you could prove the same thing about b, what can you conclude?
 
  • #7
Oh, so this says that a is divisible by (ab/m) and for b we would have mb=d(ab) so b=d(ab/m) so b is divisible by (ab/m) too. therefore (ab/m) is a common divisor of a and b. But how do we know that ab|ma? where does that come from?
 
  • #8
Look at your hypothesis: m is a common multiple of a and b; a trivially divides a. So what can you say about ma and ab?
 
  • #9
since b|m then ab|ma and since a|m then ab|mb. This leads to a is divisible by (ab/m) because ma=k(ab) and so on. I believe I finally got it?
 
  • #10
Yes. Don't forget to add that ab is divisible by m by hypothesis.
 
  • #11
Great. Thanks for all your time and patience! I really appreciate the help.
 

FAQ: Can you prove that ab/m is a common divisor of a and b?

What is a common multiples proof?

A common multiples proof is a method in mathematics to show that two or more numbers have a common multiple. It involves finding the multiples of each number and identifying the common multiples.

How do you find the common multiples of two numbers?

To find the common multiples of two numbers, you can list out the multiples of each number and identify the numbers that appear in both lists. Another method is to find the least common multiple (LCM) of the two numbers.

Why is a common multiples proof important?

A common multiples proof is important because it helps us understand the relationship between numbers and identify key patterns. It is also used in various mathematical concepts such as finding equivalent fractions and simplifying fractions.

Can a common multiples proof be used for more than two numbers?

Yes, a common multiples proof can be used for any number of numbers. The method remains the same - finding the multiples of each number and identifying the common multiples.

What are some real-life applications of common multiples proofs?

Common multiples proofs can be used in various real-life situations, such as finding the best time to schedule a meeting when multiple people are available, determining the least amount of materials needed for a project, and calculating the number of guests that can be seated at different sized tables.

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