how can I find Integer solutions for the following equation:
$$900 X \, Y + 210 Y + 30 X - 3 \times 10^{100} + 7 = 0$$
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I tried to know if I can find x and y are both integers for two variables equations without search factors. – Kumar Aug 15 '14 at 18:15
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This is equivalent to $(30X+7)(30Y+1)=3\times 10^{100}$. That should quickly yield the answer. – André Nicolas Aug 15 '14 at 18:15
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Thanks @AndréNicolas, but how I can find the integer solutions? – Kumar Aug 15 '14 at 18:16
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2There aren't any. The left side is never divisible by $3$. – André Nicolas Aug 15 '14 at 18:17
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How you know that "The left side is never divisible by 3"? – Kumar Aug 15 '14 at 18:20
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if (30X+7)(30Y+1) = 50437, then what are the integer solutions? – Kumar Aug 15 '14 at 18:25
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Same method as I described in you prior question. – Bill Dubuque Aug 16 '14 at 00:23
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the left side of the equation is odd and therefore cannot be 0 – miracle173 Jan 07 '21 at 00:05
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Hint: $(30 X + 7)(30 Y +1) = \ldots$
Robert Israel
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if (30X+7)(30Y+1) = 50437, then what are the integer solutions? any way to solve without searching factors? – Kumar Aug 15 '14 at 18:26
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What do you mean by "without searching factors"? A solution provides a factorization of $50437$ where one factor is congruent to $7 \mod 30$ and the other to $1 \mod 30$. – Robert Israel Aug 15 '14 at 18:59