Final answer:
The integer n that satisfies both the GCF of n and 18 being 6 and the LCM of n and 9 being 126 is 42.
Step-by-step explanation:
The student is asking for help with a number theory problem involving Greatest Common Factor (GCF) and Least Common Multiple (LCM). To find the integer n for which the GCF of n and 18 is 6 and the LCM of n and 9 is 126, we need to consider the prime factorization of the given numbers. The prime factorization of 18 is 2×93, and for 9, it is 3× 3. Since the GCF of n and 18 is 6, n must include the factors 2 and 3, but no additional factors of 3 since 6 contains only one 3. The LCM of n and 9 is 126, which factors as 2×93× 7. This means n must include a factor of 7, which 9 does not provide, and 2×93 from the LCM of 9 to be consistent with the GCF.
Therefore, n must be 42 to include the necessary factors of 2, 3, and 7, ensuring that the GCF with 18 is 6 (2×93) and the LCM with 9 is 126 (2×(3× 3)× 7).