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Find the missing digit so that it becomes divisible by 6 for
a. 45?673
b. 34562?
Solution:
a. Consider the given number
Since the last digit of a given number is an odd number, then it is not divisible by 6. A number is divisible by 6 if it is both divisible by 2 and 3. All even numbers are divisible by 2. There's nothing that we can do in order to become divisible by 6 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 6.
b. Consider the given number
A number is divisible by 6 if it is both divisible by 2 and 3. In short, an even number that is divisible by 3. Add all the digits of the given number as follows
Since 20 is not a multiple of 3, then we need to add a number so that it becomes a multiple of 3. So, 20 + 1 = 21. We can add also 4 (1 + 3) so that 20 + 4 = 24. We can add also 7 (1 + 3 + 3) so that 20 + 7 = 27. 7 is the highest digit that we can use because 7 + 3 = 10 will be a two digit number and we need to use only one digit to fill up the missing digit. The numbers 1, 4, and 7 are the right digits to fill up the missing digit in order to become the given number divisible by 3.
Since we want a given number to be divisible by 6, then we have to choose 4 as a digit because the missing digit is the last digit. The last digit must be an even number so that the given number becomes divisible by 6. Therefore, the possible number is 345624.