Solution to exercise at 17:30 The reason squaring 5 consecutively gives a solution (the orange one) to the x^2 ≡ x mod 10^10 automagically is because firstly, that’s one of the easiest way to find 0 in mod 5^10. The reason it also gives the solution to 1 mod 2^10 is because 5 ≡ 0 mod 5 ≡ 1 mod 2 and notice that by squaring this solution of x ≡ 0 mod 5 ≡ 1 mod 2, you actually get a solution of y ≡ 0^2 mod 5^2 ≡ 1^2 mod 2^2. By repeating this it not hard to see that 5^(2^k) solves the equation z ≡ 0 mod 5^(k+1) ≡ 1 mod 2^(k+1). However for 6 we have 6 ≡ 1 mod 5 ≡ 0 mod 2 and same method fails immediately, but one can check that squaring 6 gives a solution to x^5 ≡ x mod 10^10 eventually for pretty much the same reason. Key: binomial coefficients
Solution to exercise at 17:30
The reason squaring 5 consecutively gives a solution (the orange one) to the x^2 ≡ x mod 10^10 automagically is because firstly, that’s one of the easiest way to find 0 in mod 5^10. The reason it also gives the solution to 1 mod 2^10 is because 5 ≡ 0 mod 5 ≡ 1 mod 2 and notice that by squaring this solution of x ≡ 0 mod 5 ≡ 1 mod 2, you actually get a solution of y ≡ 0^2 mod 5^2 ≡ 1^2 mod 2^2. By repeating this it not hard to see that 5^(2^k) solves the equation z ≡ 0 mod 5^(k+1) ≡ 1 mod 2^(k+1). However for 6 we have 6 ≡ 1 mod 5 ≡ 0 mod 2 and same method fails immediately, but one can check that squaring 6 gives a solution to x^5 ≡ x mod 10^10 eventually for pretty much the same reason.
Key: binomial coefficients
Richards thanks for your lectures. Is the series Theory Of Numbers completed? Or will you make more videos on these Theory Of Numbers?
Sunzi is the same person who wrote the The Art of War (under the name of Sun Tzu).
Thanks again for professor's wonderful maths lectures.
According to wikipedia en.wikipedia.org/wiki/Sunzi_Suanjing they are two different people with the same name.
@@richarde.borcherds7998 You are right. I got this wrong all these years. Thanks for clarification.
Thankyou
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