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Because some numbers have a lot of factors. Consider 2520. The factor pairs (2520,1)(504,5)(360,7)(315,8)(280,9)(72,35)(63,40)(56,45) all are relatively prime, which means that they have a GCF of 1. Whenever a pair of numbers has a GCF of 1, the LCM is their product, which in this case is 2520. So here are 8 distinct sets of numbers with the same GCF and LCM. It happens more frequently the more factors you have.

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Q: Why is it important to find the greatest common factor?

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There cannot be a greatest common factor (GCF) of just one number. To be common there need to be at least two numbers. If you find all the factors of two or more numbers, and you find some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.

There cannot be a greatest common factor of just one number. To be common there need to be at least two numbers. If you find all the factors of two or more numbers, and you find some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.

There is always a common factor. If there are no common prime factors, the GCF is 1.

the greatest common factor is 1 because it's the only factor of 10 that 17 can be divided by without a remainder.

There cannot be a greatest common factor (GCF) of just one number. To be common there need to be at least two numbers. If you find all the factors of two or more numbers, and you find some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.

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You do a factor rainbow to find a prime factorization. You compare prime factorizations to find a greatest common factor.

The Greatest Common Factor (GCF) is: 12

The greatest common factor (GCF) is 2.

The greatest common factor (GCF) is 4.

The greatest common factor (GCF) is 3.

The greatest common factor (GCF) is 5.

The only factor they have in common is 1. It has to be the greatest.

There is no Greatest Common Factor (GCF) for a single number. The Greatest Common Factor (GCF) is the largest factor common to two or more given numbers.

There's no such thing as the greatest common factor of "a number". "Common" means "same for both". There can't possibly be anything 'common' until you have at least two separate numbers.

greatest common factor of 35 and 56 = 7

The Greatest Common Factor of 45, 65 is 5.

the greatest common factor of 8a3b2 and 12ab4 is 24a3b4