the "roundness of an integer greater than 1 is the sum of the exponenys of the prime number. for example, 20= 2to the squared multiplied by 5 to the 1st power,so 20 has a roundness of 3. what is the roundness of 1,000,000?
its 12
12
bru that was so easy from aops
12
To find the roundness of 1,000,000, we need to determine the exponent of each prime number in its prime factorization. The prime factorization of a number represents the product of its prime factors. Let's break down the steps to find the roundness of 1,000,000:
Step 1: Prime Factorization of 1,000,000
To find the prime factorization of 1,000,000, we start by dividing it by the smallest prime number, which is 2:
1,000,000 ÷ 2 = 500,000
Again, divide the quotient by 2:
500,000 ÷ 2 = 250,000
Repeat this process until we can no longer divide by 2:
250,000 ÷ 2 = 125,000
125,000 ÷ 2 = 62,500
62,500 ÷ 2 = 31,250
31,250 ÷ 2 = 15,625
Now, let's divide by the next prime number, 5:
15,625 ÷ 5 = 3,125
3,125 ÷ 5 = 625
Since 625 is a perfect square, let's divide by its square root, which is 25:
625 ÷ 5 = 125
We are now left with 125, which is divisible by the prime number 5:
125 ÷ 5 = 25
Finally, 25 is a perfect square, so we divide by the square root of 25, which is 5:
25 ÷ 5 = 5
Step 2: Exponent Calculation
Now that we have the prime factorization of 1,000,000, which is 2^6 × 5^6, we can obtain the exponent values.
The exponent for the prime number 2 is 6, and the exponent for the prime number 5 is also 6.
Step 3: Roundness Calculation
To calculate the roundness, we add the exponents of the prime numbers. In this case, the roundness of 1,000,000 is:
6 (exponent of 2) + 6 (exponent of 5) = 12
Therefore, the roundness of 1,000,000 is 12.