suppose that a and b are positive integers both divisible by 19
Mathematics

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suppose that a and b are positive integers both divisible by 19. is it possible to find intergers x and y such that 0<ax+by<19? Why or why not?
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Here is what you are given:
a and b are positive integers that are both divisible by 19.
x and y must also be integers.
The smallest number possible for a and b is 19. Suppose both a and b are 19.
Then you would need to find integers for x and y so that 0<19x+19y<19
If both x and y are positive integers, then all sums produced by adding 19x and 19y would be greater than 19.
If both x and y are negative integers, then all sums produced by adding 19x and 19y would be less than 0.
So we are left to assume that either x or y is negative, and the other variable is positive. It does not matter if x or y is negative, it only matters that only one of them is negative.
Case where absolute values of x and y are equal:
If x = 1 and y = 1, then we would have the following:
0 < 19(1)+19(1) <19 which is equivalent to 0 < 0 < 19, which is not true.
This case shows why the absolute values of x and y cannot be equal. Even if the absolute values of x and y are greater than 1 (while still being equal to each other), we would end up with the same result.
Case where absolute value of y is greater than the absolute value of x:
If x = 1 and y = 2, then we would have the following:
0 < 19(1)+19(2) <19 which is equivalent to 0 < 19 < 19, which is not true.
The case gets worse if the absolute value of y becomes increasingly greater than the absolute value of x:
If x = 1 and y = 3, then we would have the following:
0 < 19(1)+19(3) <19 which is equivalent to 0 < 38 < 19, which is not true.
The case is similar in the opposite direction, where the absolute value of x becomes increasingly greater than the absolute value of y:
Case where absolute value of y is greater than the absolute value of x:
If x = 2 and y = 1, then we would have the following:
0 < 19(2)+19(1) <19 which is equivalent to 0 < 19 < 19, which is not true.
If x = 3 and y = 1, then we would have the following:
0 < 19(3)+19(1) <19 which is equivalent to 0 < 38 < 19, which is not true.
If your class is not a proofs class, then this is enough of an explanation to show why it is not possible.
I hope this helps you move forward on the problem! Let me know if you need any more clarification!Secure Information
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