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If $p$ and $q$ are distinct prime numbers, then there are integers $x_0$ and $y_0$ such that $1 = px_0+qy_0$. Determine the maximum value of $b-a$, where $a$ and $b$ are positive integers with the following property: If $a \leq t \leq b$, and $t$ is an integer, then there are integers $x$ and $y$ with $0 \leq x \leq q-1$ and $0 \leq y \leq p-1$ such that $t = px + qy$.

The first statement is obvious. It follows from Bezout's identity. Also if we want to determine the maximum value of $b-a$ it needs to be for all distinct primes $p,q$, so how do we ensure that?

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  • $\begingroup$ By definition of the question the answer is technically $a = -\infty$, $b = \infty$, $b - a = \infty$ to create the maximum value. The question is poorly worded. $\endgroup$ – Dane Bouchie Jun 20 '16 at 15:45
  • $\begingroup$ @DaneBouchie So you are saying any integer $t$ can be represented in that form? $\endgroup$ – Puzzled417 Jun 20 '16 at 15:46
  • $\begingroup$ I'm just pointing out the question (I'm guessing you were given) is poorly worded. It asks for the maximum range. I think the meant the minimum value of $b-a$, for all $t$ such that ... etc. $\endgroup$ – Dane Bouchie Jun 20 '16 at 15:49
  • $\begingroup$ @DaneBouchie It says determine the maximum possible value of $b-a$ with the following property... so the answer isn't $\infty$. $\endgroup$ – Puzzled417 Jun 20 '16 at 15:50
  • $\begingroup$ Well as long as $t$ is some positive integer, we simply pick the largest values such that $a <= t <= b$. Therefore we choose $a = 0$ and $b = \infty$ to get $b-a = \infty$. $\infty$ is the maximum possible value of $b-a$. (Corrected that $a$ needs to be a positive integer) $\endgroup$ – Dane Bouchie Jun 20 '16 at 15:53
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If $t<p$ and $t<q$, then you can't have $x\geq0$ and $y\geq0$ with $t=px+qy$.
So $a\geq\min(p,q)$.
And the highest possible value for $t$ is $p(q-1)+q(p-1)$,
so $b\leq2pq-p-q$.
There are probably tighter bounds than that though.
If $p=3$ and $q=5$, the possible numbers are: $$0,3,6,9,12;5,8,11,14,17;10,13,16,19,22\\ =0,3,5,6,8,9,10,11,12,13,14,16,17,19,22$$ so the largest value of $b-a$ comes for $a=8$ and $b=14$.

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  • $\begingroup$ Try $2\&3$, $2\&5$, $2\&7$, $3\&5$, $3\&7$,$5\&7$ until you find a pattern for $a$ or $b$ or $b-a$ $\endgroup$ – Empy2 Jun 20 '16 at 18:14
  • $\begingroup$ I am not seeing a pattern. Can you tell me how to solve this? It looks symmetric. $\endgroup$ – Puzzled417 Jun 20 '16 at 18:22
  • $\begingroup$ I get $0,2,3,4,5,7$ for $2,3$ and $0,2,4,5,6,7,8,9,11,13$ for $2,5$. $\endgroup$ – Puzzled417 Jun 20 '16 at 18:38
  • $\begingroup$ Draw up a table with five columns p,q,a,b,b-a. The first row, from your results, is 2,3,2,5,3. When you have a few rows, try to find a connection between p,q, and the other columns. $\endgroup$ – Empy2 Jun 21 '16 at 0:26

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