If the number 321a9246b is divisible by 72, then find the value of a
2
−b
2
.

For a number to be divisible by 72, it must be completely divisible by both 8 and 9 (since 8 and 9 are co-prime factors of 72).
Step 1: Divisibility by 8
A number is divisible by 8 if its last three digits form a number divisible by 8.
The last three digits of 321a9246b are 46b.
Let's check for divisibility:

8
46b
​

460=8×57+4. So, the remainder when 460 is divided by 8 is 4.
To make it perfectly divisible, 40+b must be divisible by 8. Thus, b=4 (since 464 is divisible by 8).

Step 2: Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Sum of digits of 321a92464:

3+2+1+a+9+2+4+6+4=31+a
For (31+a) to be divisible by 9, the nearest multiple of 9 greater than 31 is 36.

31+a=36⟹a=5
Step 3: Calculate a
2
−b
2


a
2
−b
2
=5
2
−4
2
=25−16=9