Answer Key – Why GCF/LCM methods work
1. Why does the method of looking at the smallest amount of each pair of primes (obtained from the
prime factorizations) work for finding the GCF?
If you are looking at the greatest number that divides two quantities (call them A & B), then you
can just look at each of the respective prime factorizations. Why? Since each number is completely
composed of these prime factors, then the greatest number that divide both must also be completely
composed of a combination of the prime factors that A and B have in common. If a prime number is
not included in either A or B’s prime factorization, then a quantity that contains that prime number
as a prime factor will not divide either A or B. Thus, the GCF(A,B) will be completely determined by
the prime factorization of A and B.
Now why, when we look at each prime from the prime factorizations, do we pick the smallest amount
of each (upon comparison of the prime factorizations)? Well, it can’t be the largest amount. The
largest amount would not divide the quantity that contains the smaller amount. Let’s look at an
example to make it more clear. Suppose that you are given two quantities, A = 22 × 32 × 5 × 74 and
B = 23 × 3 × 5 × 7 × 132 .
Now, let’s suppose that you think that the value of 23 will be part of the GCF(A,B). If you conjecture
this, then what you are saying is that 23 divides the quantity of A. This is clearly false because 23 does
not go into 22 evenly. Another way to see this is to realize that you cannot multiply anything by 23
to get 22 (remember that we are only working with exponents that are positive or zero). Likewise, 23
does not go into 32 , 5, or 74 evenly. Therefore, 23 does not divide A. So, 23 is not going to be in the
GCF(A,B). A similar argument can be made for any value greater than 23 .
So either 22 or 21 is in the GCF(A,B). Well, we know that 21 divides both A and B. However, is it the
largest such quantity (remember that we are after the greatest common factor)? No, because we know
that 22 divides both A and B. The value of 22 will be a part of the GCF(A,B), since we know that we
need at least 1 two to be part of the GCF (it is a prime factor of both). Similar arguments can be made
for why the smaller of the remaining primes must part of the GCF(A,B). Thus, to find the GCF(A,B),
you compare prime factors obtained from the prime factorization and include the smallest amount of
each prime factor in the GCF(A,B). If the amount of prime factors is equal, then you include that
amount in the GCF.
2. Why does the method of looking at the greatest amount of each pair of primes (obtained from the
prime factorizations) work for finding the LCM?
Before we find the least common multiple of two quantities (let’s call them A & B), let’s first look at
the properties of some common multiples of A & B. Well, A · B is a common multiple of both A and
B. Thus, other common multiples of A & B take on the form 2AB, 3AB, etc. So, the biggest that
LCM(A, B) can be is A · B. Is it necessarily the smallest such common multiple? Let’s look at an
example. Let A = 12 and B = 8. Looking at A · B we get 12 · 8 = 96. Is 96 the LCM (it is in fact
a common multiple of both but is it the smallest such multiple)? Well, no because 8 · 3 = 24 and
12 · 2 = 24. The LCM of 8 and 12 is 24, not 96.
So, when does A · B translate into the LCM? This occurs when A & B have no factors in common
(other than 1 – in such a case we say that A & B are relatively prime). Take for example 15 &
8. The first multiple that these two quantities have in common is 15 · 8 = 120 and 8 · 15 = 120 (you
should verify this). Now listing all the factors of both, you should see that 15 & 8 have no factors
(other than 1) in common. Now look at each of their respective prime factorizations. You should see
that the prime factorizations don’t have any prime factors in common. So, to obtain the LCM of these
quantities, you would simply multiply all the prime factors together giving the value of 120. In the
case of 8 & 12 from above, if you look at the prime factorizations, you will see that they do indeed
have prime factors in common. Thus the LCM of those two quantities will be some combination (but
not all of) the prime factors obtained from the prime factorizations.
Okay, so now we know that we can just make a combination of some of the prime factors that are
obtained from the prime factorizations. So, we compare the prime factors and pick the larger amount.
But why pick the largest amount? Well, it can’t be the smallest amount. We know from the first
part of this assignment that picking the smallest amount gives us the GCF. So, the value has to be
quantities that are equal to or larger than the largest amount of primes (when each pair of primes is
compared).
Let’s look the same example as used in the GCF argument to make it more clear. Suppose that you
are given two quantities, A = 22 × 32 × 5 × 74 and B = 23 × 3 × 5 × 7 × 132 .
Now, we know that 22 is not part of the LCM. For one thing, it is part of the GCF. More importantly
though, the value of 22 is not part of the multiple of B, specifically it is not a multiple of 23 (which
is part of the quantity B). Recall, that if it is to be part of the LCM, it must be a common multiple
of both quantities. So, why is 22 not a multiple of 23 ? Well, there is nothing you can multiply 23 by
to obtain an answer of 22 (remember that we are only working with exponents that are positive or 0).
Since it is not a multiple of 23 (or any prime factor of the prime factorization), it is not part of the
multiple of B. A similar argument can be made for why 21 is not part of the LCM.
So, we are left with deciding if 23 will be part of the LCM or if a larger amount of 2’s is needed. Well,
any value larger than 23 is part of a common multiple of A and B. Specifically, any value that is larger
than 23 is a multiple of both 22 and 23 . Why? Well, because we can find values of x and y such that
22 · x = 2c , c > 3 and 23 · y = 2d , d > 3 are true statements. However, if you pick any amount that is
greater than 23 , you are not picking the smallest multiple that is common to both quantities A and B.
Why? Because 23 itself is both a multiple of 22 and 23 , specifically 22 · 21 = 23 and 23 · 20 = 23 . Since
we’ve already shown that 22 can’t be part of the LCM, 23 is the smallest amount. Therefore, it will
be part of the Least Common Multiple of A and B (LCM(A,B)).
Similar arguments can be made for why the larger of the remaining primes must be part of the
LCM(A,B). Thus, to find the LCM(A,B), you compare prime factors obtained from the prime factorization and include the greatest amount of each prime factor in the LCM(A,B).