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have changed your equation from 19x=29 mod16 to 19x=16 mod29 as it is impossible to have 29 mod 16 in modular arithmetic/it makes no sense.
Anyway, any modular equation a=b mod m can be rewritten as a-b=km where k is an integer. or in other words, m divides a-b without remainder. Applying this to the equation above we get (19x-16)/29=k. Now just stick it into a calculator and replace x with integers until the equation gives out an integer, in this case x turns out to be 10.
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