SOLUTION: Summing the digits. Find the sum of the digits in the standard form of the number 2^2005 * 5^2007.

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Question 217791: Summing the digits. Find the sum of the digits in the standard form of the number 2^2005 * 5^2007.

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
No wants to figure out 2%5E2005 or 5%5E2007! And even if you have a calculator which does not choke on these, it is unlikely that it will show you all the digits of the results. So there must be some "trick" to this.

What numbers are easy to raise to high powers? Answer: 0, 1, ... and 10! 10 raised to a power is a 1 followed by as many zeros as the exponent. For example 10%5E25 is a 1 followed by 25 zeros. So the "trick" to this problem is to change as much of the expression as possible to some power of 10. Since we have a lot of 2's and 5's as factors this will not be hard. We have 2005 2's and 2007 5's multiplied together. If we pair all 2005 2's with 2005 of the 2007 5's, we end up the 2005 10's with 2 5's left over:
2%5E2005+%2A+5%5E2007+=+10%5E2005+%2A5%2A5+=+25%2A10%5E2005
So we have 25 times a 1 followed by 2005 zeros. This results in a 25 followed by 2005 zeros. Those 2005 zeros are pretty easy to add up! So the sum of all 27 digits is 2+5+0+.....+0 = 7!