If the 8-digit number 63k4021p is divisible by 72, then find the value of (5k−3p).

Since the number 63k4021p is divisible by 72, it must be completely divisible by both 8 and 9 (as 72=8×9, where 8 and 9 are co-prime).
Step 1: Divisibility by 8
A number is divisible by 8 if its last three digits are divisible by 8.
The last three digits are 21p.
Dividing 21p by 8:

8
210+p
​
=
8
208+2+p
​

For (2+p) to be divisible by 8, where p is a single digit (0≤p≤9), p must be 6.
Thus, p=6.

Step 2: Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Sum of digits = 6+3+k+4+0+2+1+p
Substituting p=6:

Sum=6+3+k+4+0+2+1+6=22+k
For (22+k) to be a multiple of 9, the nearest multiple greater than or equal to 22 is 27.

22+k=27⟹k=5
Step 3: Evaluate the required value

5k−3p=5(5)−3(6)=25−18=7