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Dollydaggerxo
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Homework Statement
Prove that 10^n leaves remainder 1 after dividing by 9.
The Attempt at a Solution
There is an integer K, such that 10^n = 9k + 1
Where do i go from here if I want to do it just directly?
The concept is based on the mathematical property that the remainder of dividing any power of 10 (10^n) by 9 is always 1. This can be proven through various mathematical methods such as induction, modular arithmetic, and divisibility rules.
Proving this concept is important because it helps us understand the behavior of numbers when divided by 9. It also has practical applications in fields such as cryptography, computer science, and engineering.
Yes, this concept can be extended to any number that is one less than a multiple of 9 (e.g. 11, 1001, 1000001). This is because all these numbers can be expressed as 9n+1, where n is any positive integer.
One example is the use of checksums in computer systems. In order to detect errors in transmitted data, a checksum is calculated by dividing the sum of all digits in a number by 9. If the remainder is 1, it is likely that the data has been transmitted correctly.
Induction is a mathematical proof technique that involves proving a statement or property for all natural numbers. To prove that 10^n leaves remainder 1 when divided by 9, we can show that it is true for n=1, assume it is true for n=k, and then prove that it is also true for n=k+1. By doing this, we can show that the statement holds for all natural numbers, including 10^n.