The greatest common divisor of  and  is 10. This means that the prime factorizations of  and  must both contain a 2 and a 5. 

The greatest common divisor of  and is 9. This means that the prime factorizations of  and  must both contain two 3's.

The greatest common divisor of  and  is 8. This means that the prime factorizations of  and must both contain three 2's.

Thus:

We substitute these equalities into the given expression and simplify.

Since  and  are two-digit integers (equal to  and respectively), we must have  and . Any other factor values for or will produce three-digit integers (or greater).

is equal to , so  could be either 1 or 2. 

Therefore:

or 

Greatest common factor is a common factor shared by two or more numbers. Both numbers are even, so let's divide both numbers by two. We get . These are prime numbers (factors of one and itsef) in which we are done. Anytime we have two prime numbers or one prime and one composite number, we are finished. So the greatest common factor is .

Greatest common factor is a common factor shared by two or more numbers.  is a multiple of , so let's divide  for both numbers. We get . We are finished as these are the basic numbers. So the greatest common factor is .

Greatest common factor is a common factor shared by two or more numbers.  is a prime number.  is a composite number. Since we have a prime and composite number, the greatest common factor is . 

Greatest common factor is a common factor shared by two or more numbers. Both numbers are even, so let's divide two for both numbers. We get .  We have  one prime and one composite number, so we are finished. The greatest common factor is .

Greatest common factor is a common factor shared by two or more numbers. If you know the divisibility rule of  (sum of digits are divisible by ), then the answer is just  as the quotient is . We have a prime and composite number. However, if you don't and only know the divisibility rule of , then we can divide both numbers by  to get . We do it once more to get . Since we divided twice by , we multiply these factors and this is our greatest common factor of . Our answer is .

Greatest common factor is a common factor shared by two or more numbers. If you know divisibility rule of , then this is the answer. However, this isn't easy to spot, so we will do process of elimination. The numbers are odd and if we have even factors, we never generate odd numbers so  is wrong. Next, check divisibility rule of . The digits of  add to  which isn't divisible by  so  is wrong. Next, let's divide  into . We get a decimal value and that's wrong since if we consider  to be a multiple of , it should be a whole number and not a decimal. Finally, by dividing  and , it's also . This is our answer. To find out if  is divisible by , just add the outside digits and match the middle one. Since it does,  is divisible by .

These two numbers are definitely divisible by . When we divide both numbers by , we get  and  remaining. Since we have a combination of a prime and composite number, then we can't find any more factors. Our answer is .

Let's do some divisibility rules. For , the sum of the digits must be divisible by .

We have:

 They are both good so when we divide both numbers by , we get . Lastly they are both divisible by . So we multiply both factors to get an answer of .

Greatest common factor involves all the numbers in the set. Even though three of the numbers are divisble by ,  isn't. The only factor that satisfies all the numbers is .