What powers of two prime numbers will give a product of 1000...000(this sign --> { below the 1000...000 and below this { telling 2012 zeros )
Please show your solution and how you did it. Thank you so much.
I find your explanation obscure, but if you mean a "1" followed by 2012 zeroes, then
you have 10^2012 = 2^2012 * 5^2012
To find the powers of two prime numbers that will give a product of 1000...000 (2012 zeros), we need to determine the prime factors of this large number.
The number 1000...000 (2012 zeros) can be written as 2^(2012).
Now, let's write 2012 in binary format: 11111011000.
From this binary representation, we can see that 2012 = 2^1 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^9.
Therefore, the prime factorization of 1000...000 (2012 zeros) is 2^(2^1) * 2^(2^3) * 2^(2^4) * 2^(2^5) * 2^(2^6) * 2^(2^7) * 2^(2^9).
So, the powers of two prime numbers that will give a product of 1000...000 (2012 zeros) are:
2^2, 2^8, 2^16, 2^32, 2^64, 2^128, and 2^512.
Thus, these are the powers of two prime numbers that will give a product of 1000...000 (2012 zeros):
2^2, 2^8, 2^16, 2^32, 2^64, 2^128, and 2^512.
To find the powers of two prime numbers that will give a product of 1000...000 (2012 zeros), we need to factorize this number.
Notice that a product of 1000...000 (2012 zeros) can be written as 10^(2012). This is because the number has a 1 followed by 2012 zeros, which is equivalent to the number 1 followed by 2012 zeros.
Now, let's factorize 10^(2012). We can express 10 as 2 * 5. So, we have:
10^(2012) = (2 * 5)^(2012)
Using the property of exponents, we can rewrite this equation as:
10^(2012) = 2^(2012) * 5^(2012)
Now, let's focus on the exponents of 2 and 5.
We know that 2 is a prime number, but 5 is not a prime number. However, we can further factorize 5 as 5 = 5^1.
So, our equation becomes:
10^(2012) = 2^(2012) * (5^1)^(2012)
Using the property of exponents, we can simplify this further:
10^(2012) = 2^(2012) * 5^(2012)
Now, we can see that the powers of the prime numbers 2 and 5 that will give a product of 1000...000 are 2^(2012) and 5^(2012).
Therefore, the answer is:
2^(2012) * 5^(2012)