7. Let a, b, c € Z be integers with a and b not both 0. Let d = gcd(a, b). (a) Prove that there exist x, y € Z such that if and only if d divides c. (b) Suppose there exist xo, yo € Z such that axo + byo Show that for every k EZ, the numbers x = xo + ax + by = C kb d are integers and ax + by = c. (c) Suppose still that xo, yo € Z satisfy x = xo + and = C. and axo+byo Show that if x, y EZ satisfies the equation ax + by = c, then ka kb d d y = yo = C. y = yo ka d for some k E Z. (d) Use the results from parts (a)-(c) to explain why the equation = 18x + 42y has integer solutions, and find all integer solutions x, y € Z. €30

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