<PRE><B>Question 1121103</B>
&lt;br&gt;
The amount spent on salmon is 7x; the amount spent on trout is 5y.  The total amount spent is 77:&lt;br&gt;
{{{7x+5y = 77}}}&lt;br&gt;
Very informally....&lt;br&gt;
Since x and y are positive integers, 7x is a multiple of 7.  Then, since 77 is a multiple of 7, 5y must also be a multiple of 7.&lt;br&gt;
Since 5 is not a multiple of 7, y must be.  So the possible values of y are 7, 14, 21, ....&lt;br&gt;
But y has to be less than 77/5 = 15.4; so the only two possible values for y are 7 and 14.&lt;br&gt;
Answer: The two possible values for y are 7 and 14.&lt;br&gt;
The same solution, using formal mathematics....&lt;br&gt;
{{{7x+5y = 77}}}
{{{5y = 77-7x}}}
{{{5y = 7(11-x)}}}&lt;br&gt;
Since 5 is not a multiple of 7, y must be.&lt;br&gt;
Then, as above, the constraints on the value of y lead to exactly two possible values of y -- 7 and 14.
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