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5 October, 05:47

Find integers x and y such that 115x + 30y = gcd (431, 29)

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  1. 5 October, 07:05
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    There are none.

    Step-by-step explanation:

    gcd (431,29) = 1

    The reasons I came to this conclusion is because 29 is prime.

    29 is not a factor of 431 so we are done.

    So now we want to find (x, y) such that 115x+30y=1.

    I'm going to use Euclidean's Algorithm.

    115=30 (3) + 25

    30=25 (1) + 5

    25=5 (5)

    So we know we are done when we get the remainder is 0 and I like to look at the line before the remainder 0 line to see the greatest common divisor or 115 and 30 is 5.

    So 115x+30y=5 has integer solutions (x, y) where d=5 is the smallest possible positive such that 115x+30y=d will have integer solutions (x, y).

    So since 1 is smaller than 5 and we are trying to solve 115x+30y=1 for integer solutions (x, y), there there is none.

    Furthermore, 115x+30y=1 can be written as 5 (23x+6y) = 1 and we know that 5 is not a factor of 1.
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