Question 1187121
<br>
{{{S = 7a+b+7c}}}  [1]
{{{S = 6a+8b+6c}}}  [2]<br>
Subtract [2] from [1]:<br>
{{{a-7b+c = 0}}}
{{{a+c = 7b}}}
{{{b = (a+c)/7}}}  [3]<br>
Substitute [3] in [1]:<br>
{{{S = 7a+b+7c = 7(a+c)+(a+c)/7 = (50/7)(a+c) = 50((a+c)/7)}}}<br>
We know a, b, and c are integers, so S is an integer.  From the last equation, we also know that S is a multiple of 50, and (a+c) is a multiple of 7.<br>
We also know that S is between 51 and 149.  The only multiple of 50 between 51 and 149 is 100.  So<br>
{{{S = 100 = 50((a+c)/7)}}}
{{{(a+c)/7=2}}}
{{{a+c=14}}}  [4]<br>
Substitute [4] in [3]:<br>
{{{b = 14/2 = 7}}}  [5]<br>
Combine [4] and [5] to get the answer.<br>
ANSWER: a+b+c = 14+2 = 16 -- answer choice C<br>