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Fundamental Theorem of Arithemtic : Lowest Common Multiples and Diophantine Equations

Please solve the following problems:

1. Compute the following ...
2. Let Fm be the set of all integral multiples of the integer m. Prove that ...
3. Draw the graphs of the straight lines defined by the following Diophantine equations ...
4. Prove that every integer is uniquely representable as the product of a non-negative power of 2 ... and an odd number ...

*(Please see attachment for complete problems).


Solution Preview

1)There are a number of ways to compute the least common multiple. One way is by finding the prime factors of each number: (generally by using a factor tree).

25 = 5^2
30 = 2*3*5

Now we just take each prime factor raised to the highest power found above, which gives us LCM(25,30+=2*3*5^2=150

I'll do the same thing for part b:

42 = 2*3*7
49 = 7^2


2) Assume without loss of generality that m and n are both integers
fm is the set of all multiples of m
fn is the set of all multiples of n
The intersection of these sets, or f(mIn), is all numbers that are a multiple of both m and n.

LCM(m,n) is, by definition, the smallest number that is a multiple ...

Solution Summary

Lowest common multiples and diophantine equations are investigated and the details are discussed in the solution.