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Finding Missing Digit - Divisibility Rule, 12

__Category__: Arithmetic

"Published in Newark, California, USA"
Find the missing digit so that it becomes divisible by 13 for
a. 6__?__6
b. 205__?__
__Solution__:
In
finding the missing digit, this method is completely different from the
previous divisibility by other numbers because we will use the
principles of Algebra in solving for the unknown digit.
a. Consider the given number
Let x be the unknown ten's digit. The given number can written as
To
test the divisibility of a number by 13, multiply the last digit by 4 and then add it to the remaining digits. If the result is a multiple of 13,
then the given number is divisible by 13. Let's do this for the given
number as follows
Next, equate this to the first multiple of 13 which is 13, we have
Since
the answer is negative, then we cannot accept this one because we need a
positive value for the unknown digit. The multiples of 13 are 13, 26, 39, 52, 65, 78, 91, 104, and so on. We need to choose a number which is greater than 84 that is 91 in order to get a positive value of x. Let's equate the above equation to 91, we have
Since the answer is positive, then we can accept this one. We need to end this process because we want a digit that is less than 10. Therefore, the possible number is only **676**. You can check this number by using a calculator and this number is divisible by 13.
b. Consider the given number
Let x be the unknown one's digit. The given number can written as
To
test the divisibility of a number by 13, multiply the last digit by 4
and then add it to the remaining digits. If the result is a multiple of
13,
then the given number is divisible by 13. Let's do this for the given
number as follows

Next,
equate this to the multiple of 13 which is close to 205. We want a digit
that is positive, whole number, and less than 10. Let's try 208 first,
we have

Since the answer is a fraction, then we cannot accept this one. Next, try to equate the above equation to the next multiple of 13 which is 221, we have
Since the answer is positive, then we can accept this one. We need to end this process because we want a digit that is less than 10. Therefore, the possible number is only **2054**. You can check this number by using a calculator and this number is divisible by 13.