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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.
The LCM of two numbers is 2722 and their GCF is 22. The numbers are in a ratio of 9:14. Which ones of the statements below are true?
The numbers are in the ratio:
We factorize the LCM and the GCF to see what factors they have:
Since their GCD is 22 then both numbers are multiples of 2 and of 11:
Since the LCM has a factor of 4, one of the numbers must have a factor of 4. The other number can only have a factor of 2 since the GCF only has a 2. From the ratio of the numbers, it is clear that the larger number is the multiple of 4.
The LCM of these numbers is already the desired one.
Therefore, the numbers are 198 and 308.
The difference of these numbers is a multiple of 10:
Neither one of the numbers can have 5 as a divisor since 5 is not a factor in the GCF.
From the above, it is clear that the solution to this problem is unique.