Question 1169165
<br>
The other tutor tried a solution by trial and error; but an arithmetic error led to an incorrect answer.<br>
A number y is divided by another number x to get a quotient of a and a remainder of b.<br>
The same number y is then divided by x/2.  There are two possible results:<br>
(a) the quotient is 2a and the remainder is again b; or
(b) the quotient is 2a+1 and the remainder is less than b<br>
In this problem, the remainder when y is divided by x/2 is less than the remainder when it is divided by x.  That means we have the second case.<br>
y divided by x gives remainder 29:
{{{y = ax+29}}} for some integer a<br>
y divided by x/2 gives remainder 13:
{{{y = (2a+1)(x/2)+13}}}<br>
Set the two expressions for y equal to each other and solve the resulting equation:<br>
{{{ax+29 = (2a+1)(x/2)+13}}}
{{{ax+29 = ax+x/2+13}}}
{{{29 = x/2+13}}}
{{{16 = x/2}}}
{{{x = 32}}}<br>
ANSWER: x=32<br>