Question 1177011
<br>
The number of bones she has is a multiple of 8, because she could have divided the bones equally among her dogs if she had had 8 dogs instead of 7:
{{{N = 8a}}}<br>
She would have needed 2 more bones to divide them equally among her 7 dogs, so the number of bones she has is 2 less than a multiple of 7:
{{{N = 7b-2}}}<br>
So we have<br>
{{{8a = 7b-2}}}<br>
This is a Diophantine equation: two variables but only one equation and the variables must have whole number values.  Here is a standard method for solving linear Diophantine equations.<br>
(1) Solve for the variable with the smaller coefficient:<br>
{{{8a = 7b-2}}}
{{{7b-2 = 8a}}}
{{{7b = 8a+2}}}<br>
Perform the division as a quotient and a remainder:<br>
{{{7b = 7a+(a+2)}}}
{{{b = a + (a+2)/7}}}<br>
a and b are integers, so {{{(a+2)/7}}} has to be an integer.  So a can be any integer that is 2 less than a multiple of 7.<br>
POSSIBLE ANSWERS:<br><pre>
(1)               a = 5
    b = a + (a+2)/7 = 6
             N = 8a = 40
           N = 7b-2 = 40

(2)               a = 12
    b = a + (a+2)/7 = 14
             N = 8a = 96
           N = 7b-2 = 96

(3)               a = 19
    b = a + (a+2)/7 = 22
             N = 8a = 152
           N = 7b-2 = 152</pre>
And of course there is no end to the list of possible answers.  However, only the smallest few are reasonable answers in terms of number of bones and number of dogs.<br>