Since the numbers 144 and 220 both leave a remainder of 11 when divided by 'n', I started by working out what the numbers were that could be exactly divided by 'n'.
144 - 11 = 133
220 - 11 = 209
So now we know that 133 and 209 are the numbers that can be exactly divided by 'n' without leaving a remainder. Then I used trial and error to work out the value of 'n'. I knew that 'n' had to be a multiple of 133 and 220 and also that 'n' would be an odd number. This is because 133 and 209 are both odd numbers and factors of odd numbers are odd too.
1,2,3,4,5,6,8,10,12,14,16,18 or 20 could definitely not be 'n'. 133 is a multiple of 7 but 209 isn't. That eliminated 7 too as a possibility. Then I worked in order through all the numbers left until I found that 19 is an exact multiple of both numbers. If you divide the original numbers, 144 and 220, by 19 they do leave a remainder of 11.
Therefore, the value of 'n' is 19.