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okay so we're doing inverse mod numbers so start with the example 17 X is equivalent to 1 mod 43 now it's just asking what number times 17 will give you 1 when it's more 43 so to get X on its own we just take 17 over the other side so that will give us 1 over 17 mod 43 and that's just the same as saying X is equivalent to 17 to the power of negative 1 mod 43 now to work that out we use the Euclidean algorithm and the extended Euclidean algorithm so we start off with this 43 here and that will be equal to 17 times what which is 2 plus 9 and then we want to underline these two here just so you remember that those are the two that we want to bring down next time so then we bring this 17 down bring the 9 down and so that's times what number which is obviously 1 and then plus 8 underline them again okay and then bring the 9 down bring the 8 down and that's obviously times 1 plus 1 now when we get a 1 in this position that means that the GCD of 43 and 17 is 1 we can only ever find an inverse


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