Solving the GCD of 2^10 and 10!: A Number Trick?

  • Thread starter lordy12
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In summary, 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, we need to find the prime factors of both numbers and determine the common factors. 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, and 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 are 2 x 2 x 2 x
  • #1
lordy12
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large number!

1. Finding gcd(2^10,10!



Homework Equations





3. I attempted to try the Euclidean algorithm, but it would take forever. Is there a certain number trick?
 
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  • #2
gcd(a,b) is the largest number that divides both a and b. It must be a power of 2. Why did I say that?
 
  • #3
i know its a power of 2, but you still have to use the euclidean algorithm. I don't want a "guess and check process."
 
  • #4
Then think of prime factors. How many 2's are there in the factorization of 10! ?
 
  • #5
i got it. thanks
 

1. What is the GCD of 2^10 and 10!?

The GCD (Greatest Common Divisor) of 2^10 and 10! is the largest positive integer that divides both numbers without leaving any remainder.

2. How do you solve the GCD of 2^10 and 10!?

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.

3. What is the prime factorization of 2^10 and 10!?

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.

4. How do you find the common factors of 2^10 and 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.

5. What is the GCD of 2^10 and 10! using the number trick?

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.

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