Question 1195792
<br>
Let x be the number of coins in each of the original 7 bags; the total number of coins originally was then 7x.<br>
After finding an 8th bag containing 53 coins, the total number of coins is 7x+53.<br>
That total number of coins can be distributed equally into 8 bags, so 7x+53 is a multiple of 8.  So we need to find solutions in positive integers of the equation<br>
{{{8y=7x+53}}}<br>
This is a linear Diophantine equation -- a single equation with two unknowns, whose solution(s) can be found knowing that both variables have integer values.<br>
One standard method for finding the solutions is to solve the equation for one variable in terms of the other, as follows.<br>
{{{8y=7x+53}}} [1]
{{{8y=8x+(53-x)}}} [2]
{{{y=x+((53-x)/8)}}} [3]<br>
In that equation, x is a positive integer; and y has to be a positive integer.  That means (53-x)/8 is an integer.<br>
Note that we could have looked for solutions in positive integers to equation [1] itself; however, performing the steps in [2] and [3] gives us an equation for which it is much easier to find the solutions.<br>
Remember that we are trying to find the smallest value of 7x, which is the original number of coins, given that the final total number of coins, 7x+53, is greater than 200.<br><pre>
   x    y         7x    7x+53
  ----------------------------
   5  5+48/8=11   35      88
  13 13+40/8=18   91     144
  21 21+32/8=25  147     200
  29 29+24/8=32  203     256</pre>
The smallest value of 7x for which 7x+53 is greater than 200 is 203.<br>
ANSWER: The total number of coins before finding the 8th bag was 203.<br>