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Finding Missing Digit - Divisibility Rule, 11

__Category__: Arithmetic
"Published in Newark, California, USA"
Find the missing digit so that it becomes divisible by 12 for
a. 234__?__89
b. 34524__?__
__Solution__:
a. Consider the given number
A number is divisible by 12 if it is both divisible by 3 and 4. Since the last digit of a given number is an odd number, then the given number is not divisible by 12. The multiples of 12 are all even number. There's nothing that we can do in
order to become divisible by 12 since the last digit of a given number is
not an even number. You can assign any number to the missing digit but
still, the given number will never become divisible by 12.
b. Consider the given number
A number is divisible by 12 if it is both divisible by 3 and 4. Since the missing digit is the last digit, then we can assign even number digits so that the last two digit becomes divisible by 4. When you add all the digits, the sum should be a multiple of 3 so that the given number is divisible by 12. If the last digit is 0, then the last two digit becomes 40. The sum of the digits is 3 + 4 + 5 + 2 + 4 + 0 = 18. Since 18 is a multiple of 3, then 0 is the last digit. If the last digit is 4, then the last two digit becomes 44. The sum of the digits is 3 + 4 + 5 + 2 + 4 + 4 = 22. Since 22 is not a multiple of 3, then we cannot use 4 as the last digit. If the last digit is 8, then the last two digit becomes 48. The sum of the digits is 3 + 4 + 5 + 2 + 4 + 8 = 26. Since 26 is not a multiple of 3, then we cannot use also 8 as the last digit. Therefore, the possible number is **345240** only.