Find the gcd (a, b) and lcm (a, b) if a = 2^3 5^6 and b = 2^3 5^3 7^3
To find the greatest common divisor (gcd) and least common multiple (lcm) of two numbers, we can use the prime factorization method.
First, we need to find the prime factorizations of the given numbers a and b.
Prime factorization of a = 2^3 * 5^6
Prime factorization of b = 2^3 * 5^3 * 7^3
Now, we can find the gcd(a, b) by taking the smallest power of each common prime factor. In this case, the common prime factors are 2 and 5.
For the factor 2, the smaller power is 3.
For the factor 5, the smaller power is 3.
Therefore, gcd(a, b) = 2^3 * 5^3 = 1000.
Next, we can find the lcm(a, b) by taking the highest power of each prime factor. In this case, we consider all the prime factors from both numbers, which are 2, 5, and 7.
For the factor 2, the highest power is 3.
For the factor 5, the highest power is 6.
For the factor 7, the highest power is 3.
Therefore, lcm(a, b) = 2^3 * 5^6 * 7^3 = 122,640,000.