if a=2^3*3^7*5^3*11^4 and b=2^2*3^5*7^2*11*13,find the following.(leave your anwer written with exponents)a.GCD b.LCM
a. The Greatest Common Divisor is the product of prime factors that are common to both a and b.
That would be 2^2*3^5*7*11
b. The Lowest Common Multiple is the product of all prime factors multiplied the maximum amount of times they occur in either a or b.
That would be
2^3*3^7*5^3*7^2*11^4*13
thank you so much. i just figured this out right before i checked back. i got the same answer for the lcm but for the gcd i had the same except for the 7. are you sure that is supposed to be there?
Yes. Seven is an even divisor of both a and b
To find the GCD (Greatest Common Divisor) of two numbers, we need to determine the highest power of each prime factor that appears in both numbers.
For the GCD of a and b:
a = 2^3 * 3^7 * 5^3 * 11^4
b = 2^2 * 3^5 * 7^2 * 11 * 13
Now let's find the common prime factors and their highest powers:
Common prime factors:
2: Appears in both a and b, with the highest power being 2^2.
3: Appears in both a and b, with the highest power being 3^5.
11: Appears in both a and b, with the highest power being 11^1.
Other prime factors don't appear in both a and b, so we do not include them in the GCD.
Now we can write the GCD using the common prime factors and their highest powers:
GCD = 2^2 * 3^5 * 11^1 = 4 * 243 * 11 = 106884.
To find the LCM (Least Common Multiple) of two numbers, we need to determine the highest power of each prime factor that appears in either number.
For the LCM of a and b:
a = 2^3 * 3^7 * 5^3 * 11^4
b = 2^2 * 3^5 * 7^2 * 11 * 13
Now let's find the prime factors and their highest powers:
Prime factors and their highest powers:
2: Appears in a with the highest power being 2^3, and in b with the highest power being 2^2. So we take the higher power, which is 2^3.
3: Appears in a with the highest power being 3^7, and in b with the highest power being 3^5. So we take the higher power, which is 3^7.
5: Appears in a with the highest power being 5^3, and doesn't appear in b. So we take 5^3.
7: Appears in b with the highest power being 7^2, and doesn't appear in a. So we take 7^2.
11: Appears in a with the highest power being 11^4, and in b with the highest power being 11^1. So we take the higher power, which is 11^4.
13: Appears in b with the highest power being 13, and doesn't appear in a. So we take 13.
Now we can write the LCM using the prime factors and their highest powers:
LCM = 2^3 * 3^7 * 5^3 * 7^2 * 11^4 * 13 = 8 * 2187 * 125 * 49 * 14641 * 13 = 2491501045000.
Therefore, the answers are:
a. GCD = 106884
b. LCM = 2491501045000