Final answer:
The correct common multiples of 60 and 2240 from the given options are (c) 2⁶ x 3 x 5 x 7² and (g) 2⁶ x 3 x 5 x 7, because they contain all necessary prime factors with the appropriate exponents.
Step-by-step explanation:
The question is about finding common multiples of 60 and 2240 based on their prime factor decompositions. To find a common multiple, the prime factors of both numbers must be included in the product, and the exponents for shared bases should be the highest from either number.
60 = 2² x 3 x 5
2240 = 2⁶ x 5 x 7
The common multiple must have at least the factors:
2⁶ (the higher exponent for the common prime factor 2),
5 (common prime factor),
3 (unique to 60 but necessary for a common multiple),
7 (unique to 2240 but necessary for a common multiple).
Checking the options given:
- (a) 2⁴ x 3 x 5² x 7 is not a common multiple because it only has 2 raised to the 4th power, and we need at least 2 raised to the 6th power.
- (b) 2⁴ x 3 x 7 suffers from the same issue as (a); it is missing the proper exponent for 2 and does not include the factor of 5.
- (c) 2⁶ x 3 x 5 x 7² is a common multiple because it has the necessary factors, with sufficiently high exponents.
- (d) 2² x 5, (e) 2⁸ x 3 x 5² x 7, (f) 2² x 3 x 5, and (h) 2⁶ x 5 x 7 are not common multiples for various reasons, such as lacking a necessary factor or not having the proper exponents.
- (g) 2⁶ x 3 x 5 x 7 is a common multiple and contains the exact factors of both numbers with the necessary exponents.
Therefore, the correct options for common multiple of 60 and 2240 are (c) and (g).