Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

What is the greatest common divisor of 2 to the 2021 plus 2 to the 2022 and 3 to the 2021 plus 3 to the 2022?  We can find the greatest common divisor of these two numbers by finding their prime factorizations.  So for the first number, there's a common factor of 2 to the 2021 in both, and we can factor it out to get 2 to the 2021 times 3.  For the next number, we can pull out 3 to the 2021 to get 3 to the 2021 times 1 plus another 3 coming from here.  Since 1 plus 3 is 4, which is 2 squared, this prime factorization is right here, 2 squared times 3 to the 2021.  Now to find the GCDs, we just need to look at the number of powers of 2 that are shared and the number of powers of 3 that are shared.  So 2 squared is shared among both of these, and 3 to the first power is shared among both of these, so the GCD is 2 squared times 3, which is 12.

Video Summary

What is the greatest common divisor of 2 to the 2021 plus 2 to the 2022 and 3 to the 2021 plus 3 to the 2022? We can find the greatest common divisor of these two numbers by finding their prime factorizations. So for the first number, there's a common factor of 2 to the 2021 in both, and we can factor it out to get 2 to the 2021 times 3. For the next number, we can pull out 3 to the 2021 to get 3 to the 2021 times 1 plus another 3 coming from here. Since 1 plus 3 is 4, which is 2 squared, this prime factorization is right here, 2 squared times 3 to the 2021. Now to find the GCDs, we just need to look at the number of powers of 2 that are shared and the number of powers of 3 that are shared. So 2 squared is shared among both of these, and 3 to the first power is shared among both of these, so the GCD is 2 squared times 3, which is 12.

Keywords

greatest common divisor

prime factorization

GCD calculation

shared powers

mathematics