Answer: 9540

Using divisibility tests

9  5   ___   ___

9  5   ___  (0 or 5) (since divisible by 5)

9  5   ___   0 (divisible by 6, so even)

Divisible by 3 $\Rightarrow$ sum of digits a multiple of 3

9 + 5 = 14, so this digit can be 1, 4 or 7

9510, 9540 or 9570

Divisible by 4 $\Rightarrow$ last 2 digits divisible by 4

9540

Note: You can apply the divisibility tests in a different order, but some orders will take longer than others!

Finding a larger number that the number must be divisible by

The number is divisible by 3, 4, 5 and 6, so it is divisible by 60 (lowest common multiple).

The number is less than 9600. 96 = 60 + 36, so 96 is a multiple of 6, so 960 is a multiple of 60 and therefore 9600 is a multiple of 60.

Multiples of 60: 9600, 9540, 9480

Only 9540 begins 95.