Download Euclidian Algorithm

Honors Advanced Math
Another way to find the GCF
Euclidian algorithm
Consider the problem of finding the GCF of (150, 42). One way to go about this problem would
be to simply find the prime factorization of both numbers and use it to find the GCF. This
technique works well for small numbers, but for large numbers and for certain types of
polynomials, it can be far more efficient to use a more direct algorithm called Euclid’s algorithm.
Let us begin by dividing 42 into 150 (that is, dividing the smaller number into the larger one).
WE find that 42 goes into 150 three times with a remainder of 24. We will now show that GCF
(150, 42) = GCF (42, 24).
Let G = GCF (150, 42) and N = GCF (42, 24). Our goal is to show that G = N. We will do this by
showing that G ≤ N and N ≤ G are both true statements.
Since G is the GCF (150, 42), G is a factor of both 150 and 42. So 150 = k1G and 42 = k 2 G .
Since 42 divides into 150 three times with a remainder of 24, we can make the following
conclusions:
150 = 3 ⋅ 42 + 24
k1G = 3k 2 G + 24
k1G − 3k 2 G = 24
(k1 − 3k 2 )G = 24
Thus, 24 = k 3 G , where k 3 = k1 − 3k 2 , and therefore G is a also a factor of 24.
Since G is a factor of 42 and 24, G ≤ N . (Remember, N is the greatest common factor of 42 and
24). We will now show that N ≤ G as well, using essentially the same type of reasoning.
Since N is the GCF (42, 24), N is a factor of both 24 and 42. So 24 = c1N and 42 = c 2 N . Since 42
divides into 150 three times with a remainder of 24, we can make the following conclusions:
150 = 3 ⋅ 42 + 24
150 = 3c N + c€N
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150 = (3c 2 + c1 )N
Thus, 150 = c3 N , where c3 = 3c 2 + c1 , and therefore N is a also a factor of 150. Thus, N must be
N ≤ G (because G is the greatest common factor). Since G ≤ N and N ≤ G , we can conclude
that G = N .
We have shown that GCF (150, 42) = GCF (42, 24). We can use this process repeatedly. 24 goes
into 42 one time with a remainder of 18, so GCF (42, 24) = GCF (24, 18). (This is true for
exactly the same reasons that GCF (150, 42) = GCF (42, 24)). 18 goes into 24 once with a
remainder of 6, so GCF (24, 18) = GCF (18, 6). 6 goes into 18 evenly, so GCF (18, 6) = 6. To
summarize, GCF (150, 42) = GCF (42, 24) = GCF (24, 18) = GCF (18, 6) = 6.
2.4 Extra problems
page 2
Using Euclid’s algorithm
1. Use Euclid’s algorithm to find GCF (42, 60). Confirm your answer by factoring 42 and 60.
Then use your answer to find LCM (42, 60).
2. Use Euclid’s algorithm to find GCF (48443, 49949). Then use your answer to find LCM
(48443, 49949). Feel free to use your calculator to aid your calculations. (To see the strength
of Euclid’s method, try to do this one without Euclid’s algorithm. You will find that these
numbers are fairly resistant to being factored).
3. Find GCF (416021, 1142611) and LCM (416021, 1142611). (This one would be even more
awful to do without the algorithm).
4. It is important to note that while Euclid’s method gives us a quick way of finding the GCF of
two numbers, it doesn’t necessarily tell us any information about the factorization of either
number.1 Find GCF (2491, 1763) and LCM (2491, 1763). Then explain what information, if
any, this tells us about the factorization of either 2491 or 1763.
5. Euclid’s algorithm also works finding the GCF’s of polynomials. (It works for polynomials
for essentially the same reasons that it works for whole numbers). Use Euclid’s algorithm to
find GCF ( x 3 + 3x 2 − x − 3, x 2 + 7x +12 ) and LCM ( x 3 + 3x 2 − x − 3, x 2 + 7x +12 ). These
polynomials are easy to factor, so you should be able to confirm your answer for this
problem. Hint for using Euclid’s method: Remember that the lead coefficient doesn’t matter
for the GCF. Use this fact to simplify and expedite your calculations.
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6. Use Euclid’s algorithm to find GCF ( x +12x +13, x 3 − 3x 2 +15x − 26) . (This one can also
be done without Euclid’s algorithm; see if you can do it this way too).
7. Consider GCF ( x 4 + 2x 3 +12x 2 + 4 x + 20, x 4 + 3x 3 +17x 2 + 20x + 50) . This one is much
€ method. Explain why, then use the algorithm to find the GCF.
harder to do without Euclid’s
Note: Any€of the GCF problems from the previous handout (“Greatest common factor and Least
common multiple”) can be solved by Euclid’s algorithm. Feel free to use these as necessary for
further practice.
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Currently, there is no known algorithm for finding the prime factorization of large numbers in a computationally
efficient manner. This strange fact provides the foundation for the bulk of modern encryption.