Once you know the GPGCF, you’re two easy steps from finding the GCF.
[If you don’t know how, see my last post: “Recent Insight on the GCF (and GPGCF)”] That’s one of the benefits of finding the GPGCF — speed in getting the GCF! Here are the short, sweet steps:
1) Find all factors of the GPGCF, and list them from largest to smallest.
2) Starting with the largest factor and working your way down the list, test to find the first factor that goes into both numbers. The first (largest) to do so is the GCF. You can bet on it!
Example 1 (Easy): Find GCF for 30 and 42.
1st) GPGCF is 12. Factors of 12, greatest to least, are 12, 6, 4, 3 and 2.
2nd) Largest factor to go evenly into 30 and 42 is 6. So 6 is GCF.
Example 2 (Harder): Find GCF for 72 and 120.
1st) GPGCF is 48. Factors of 48, greatest to least, are 48, 24, 16, 12, 8, 6, 4, 3 and 2.
2nd) Largest factor to go evenly into 72 and 120 is 24. So 24 is GCF.
NOW TRY THESE —
For each pair:
1) Find GPGCF and say if it is the difference or smaller #.
2) List factors of GPGCF, greatest to least.
3) Find GCF.
a) 8 and 12
b) 16 and 40
c) 18 and 63
d) 56 and 140
ANSWERS:
a) 8 and 12
GPGCF = 4 (difference)
Factors of 4: 4 and 2
GCF = 4
b) 16 and 40
GPGCF = 16 (smaller #)
Factors of 16: 16, 8, 4 and 2
GCF = 8
c) 18 and 66
GPGCF = 18 (smaller #)
Factors of 18: 18, 9, 6, 3 and 2
GCF = 6
d) 56 and 76
GPGCF = 20 (difference)
Factors of 20: 20, 10, 5, 4 and 2
GCF = 4
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Comments on: "From GPGCF to GCF … in two easy steps" (4)
RE: Example 2 (Harder): Find GCF for 72 and 120.
1st) GPGCF is 48. Factors of 48, greatest to least, are 48, 24, 16, 12, 8, 6, 4, 3 and 2.
2nd) Largest factor to go evenly into 72 and 120 is 24. So 24 is GCF.
Q – is there a shortcut of knowing that 24 is the GCF other than trial and error? Because I will be trying 72/48 and 120/48, then 72/24 and 120/24.
I bookmarked your website. It is very helpful. Thank you for sharing and for being generous of our knowledge.
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Hi again.
Very good question. Thanks again for asking it!
Unfortunately, I don’t know of any quick-or-trick way to find out which of the GPGCF’s factors will be the actual GCF. So yes, you do need test them one by one, starting with the largest and working your way down the list.
Of course, when you learn tricks for divisibility (see my several blogs on divisibility), you can use many of those tricks to find out quickly whether or not a number divides evenly into another number.
And if you get fast at prime factorizing numbers, you can use prime-factor info to find the GCF directly. For example, with the number pair you mentioned, 72 and 120, prime factorizing tells you that:
72 = (2^3) x (3^2), and that
120 = (2^3) x (3) x (5)
Using that knowledge, you just grab the largest factor in common for each prime that the numbers share, which would be:
2^3 (from the 2s) and 3 (from the 3s).
Then multiply those: 2^3 x 3 = 8 x 3 = 24, and that is the GCF.
So while there’s no individual trick I know of for the specific question you raise, there are other tricks that you can use instead.
Best,
— Josh
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How to get GCF of 2 prime numbers? example 11 and 19
How to get GCF of 1 prime and 1 composite numbers? example 17 and 48
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Hi, and thanks for sharing your questions. They are good ones.
The GCF of two prime numbers is, sadly, just 1. In other words, there is no interesting GCF for two primes because if two numbers are prime, they have no common factors except 1.
For a prime and a composite, there are two possibilities. 1) If the prime number is smaller than the composite, and if it divides evenly into the composite, then the prime number is itself the GCF. An example would be 7 and 21. 7 is prime, but it does divide evenly into the composite number 21. So for this number pair, 7 is the GCF. 2) If the prime is larger than the composite, or if the prime is smaller but it does not divide evenly into the composite, then the GCF for the pair is again just 1. An example of the first case is 23 and 16. 23 is prime, but it’s larger than the composite number 16. So there’s no way that 23 could divide into 16, and therefore the GCF is just 1. An example of the second case is 7 and 16. 7 is prime and it does not divide evenly into the composite number 16. So here, too, the GCF is just 1.
I’m glad to see you thinking and asking questions. Numbers are pretty amazing when you start “scratching below the surface.”
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