A number when divided by a divisor, leaves a remainder of 24. When twice that number is divided by the same divisor, the remainder is 15. What is the divisor?

Let's just assume the number be N

and the divisor be D

we know that

$$Number= Divisor \times quotient + Remainder$$

$$N={D}\times{K}+24$$, (where k is the quotient)

When twice the number that is 2N

$${2}\times{N}= {{2}\times{D}\times{k}}+48$$       [ i.e $${2}\times{24}=48$$]

As the remainder of 2N divided by D is 15 , therefore 48 must go into the quotient in the form $$\frac{33}{D}$$

Therefore the divisor is  $$48-15= 33$$