The GCD (Greatest Common Divisor) of 2^10 and 10! is the largest positive integer that divides both numbers without leaving any remainder.
To solve the GCD of 2^10 and 10!, we need to find the prime factors of both numbers and then determine the common factors. Then, we multiply the common factors to get the GCD.
The prime factorization of 2^10 is 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2. The prime factorization of 10! is 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10.
The common factors of 2^10 and 10! are the prime factors that appear in both numbers. In this case, the common factors are 2 x 2 x 2 x 2 x 2.
The number trick to find the GCD of 2^10 and 10! is to take the smaller exponent of each prime factor that appears in both numbers. In this case, the GCD is 2^5 = 32.