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The Really Really Big Number
Letter-Equations
Letter Equations are well known phrases or facts where the key words have been replaced with the first letter of that word. These are often in the form of an equation, which contain a number, an = sign and the rest of the obscured phrase or fact.Letter-Equations
When you divide 12 by 5, the remainder is 2; it's what's left over after you have removed all the 5s from the 12. When you raise 4 to the fifth power (that is, 4^5), you multiply four by itself five times: 4x4x4x4x4, which equals 1,024.
What is the remainder when you divide 100^100 by 11?
What is the remainder when you divide 100^100 by 11?
Answer
1This one is so sneaky.
First, consider 100 divided by 11. The remainder here is 1. Now consider the remainder when 100x100 is divided by 11. Don't do it on your calculator or on paper. Rather, consider that you have one hundred hundreds, and each of them has a remainder of 1 when divided by 11. So, go through each of your hundred hundreds and divide it by 11, leaving remainder 1. Then collect up your remainders into a single hundred, and divide it by 11, leaving a remainder of 1. This process can be extended to dividing 100x100x100 by 11, and indeed, to dividing any power of 100 by 11.
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Comments
This is not a letter equation. I love this puzzle as a math teaser, but I'm not certain I agree with the logic of the answer. I don't know that you can "collect up" remainders and divide again.
For example, what is the remainder when you divide 12^2 by 11? By the puzzle's logic, I can divide each 12 by 11 and get a remainder of 1, then gather up my two 1's to get 2. But 12x12 is 144, and 11x13=143. So the remainder of (12^2)/11 is 1, not 2.
The correct logic is 100^n = 100 * (100^(n-1)) = (99 + 1) * (100 ^ (n-1)) = 99*(100^(n-1)) + 100^(n-1)
Dividing by 11, the first term has remainder zero, so only the remainder of the second term matters.
Repeat n-1 times to get down to 100 / 11. Now you can see the remainder is 1.
For example, what is the remainder when you divide 12^2 by 11? By the puzzle's logic, I can divide each 12 by 11 and get a remainder of 1, then gather up my two 1's to get 2. But 12x12 is 144, and 11x13=143. So the remainder of (12^2)/11 is 1, not 2.
The correct logic is 100^n = 100 * (100^(n-1)) = (99 + 1) * (100 ^ (n-1)) = 99*(100^(n-1)) + 100^(n-1)
Dividing by 11, the first term has remainder zero, so only the remainder of the second term matters.
Repeat n-1 times to get down to 100 / 11. Now you can see the remainder is 1.
And I just noticed this is Beverly90210's first teaser! Congrats!
Like I said above, I love this puzzle as a math teaser. Clearly, I had fun with this and spent time thinking about it. I look forward seeing what else you may come up with.
I am going to share this puzzle with my friends and co-workers. Thanks for creating it!
Like I said above, I love this puzzle as a math teaser. Clearly, I had fun with this and spent time thinking about it. I look forward seeing what else you may come up with.
I am going to share this puzzle with my friends and co-workers. Thanks for creating it!
Still playing with this. It looks like you can find a remainder by multiplying the remainders of factors. So, in your example of 100 100's, you get 100 remainder 1's, which you can multiply together to get a final remainder 1. Interesting!
For another example, look at the remainder when dividing 12x12 by 7. You get two remainder 5's from dividing each 12 by 7. Five times five = 25, and 25/7 has remainder 4. Note 12x12/7 = 144/7 = 20x7 + 4. The remainders match.
I never contemplated that before. Thanks Beverly90210!
For another example, look at the remainder when dividing 12x12 by 7. You get two remainder 5's from dividing each 12 by 7. Five times five = 25, and 25/7 has remainder 4. Note 12x12/7 = 144/7 = 20x7 + 4. The remainders match.
I never contemplated that before. Thanks Beverly90210!
your welcome snowdog
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