Question 9536
Let A be her age this year then A = 7k for some positive integer k

 A+1  is her age next year and A+1 = 4l for some positive integer l

 1 = A+1 -A

   = 4l - 7k

 We have to solve 4l - 7k = 1 in positive integers

 We write equivalently 4l -8k + k = 1

          4l - 8k = 1 -k

   Let 1-k = 4r for some integer r

 Then k  = 1 -4r

 Substitute for k and solve for l in 4l - 7k = 1

 We get l = 2-7r

 A = 7k
    = 7 -28r

 10 < A < 60

 By substituting for A and solving for r we get

 -53/28 < r < -3/28

  The only integer that satisfies this is r=-1

 Then A = 7-28r 

        = 35

 Ann is 35 this year which is a multiple of 7
next year she is 36 which is a multiple of 4

 and 35 is between 10 and 60