## Thursday, May 29, 2014

### Finding Missing Digit - Divisibility Rule, 6

Category: Arithmetic, Algebra

"Published in Newark, California, USA"

Find the missing digit so that it becomes divisible by 7 for

a. 2?4
b. 108?

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 7, double the last digit and then subtract it to the remaining digits. If the result is a multiple of 7, then the given number is divisible by 7. Let's do this for the given number as follows

Next, equate this to the first multiple of 7 which is 7, we have

Since the answer is negative, then we cannot accept this one because we need a positive value for the unknown digit. Let's equate the above equation to the next multiple of 7 which is 14, we have

Since the answer is positive, then we can accept this one. Let's equate the above equation to the next multiple of 7 which is 21, we have

Since 9 is the highest digit, then we can end this process because we want a digit that is less than 10. Therefore, the possible numbers are 224 and 294. You can check these numbers by using a calculator and these numbers are divisible by 7.

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 7, double the last digit and then subtract it to the remaining digits. If the result is a multiple of 7, then the given number is divisible by 7. Let's do this for the given number as follows

Next, equate this to the multiple of 7 which is close to 108. We want a digit that is positive, whole number, and less than 10. Let's try 98 first, we have

Since the answer is a positive whole number, then we can accept this one. Next, try to equate the above equation to the next multiple of 7 which is 105, we have

Since the answer is not a whole number, then we cannot accept this one. Next, try to equate the above equation to the next multiple of 7 which is 112, we have

Since the answer is a negative number, then we cannot accept this one also. If you will continue this process to the next multiple of 7 like 119, 126, 133 and so on, all values of x are negative. We have to assign a multiple of 7 that is less than 108. Let's try 91 first, we have

Since the answer is not a whole number, then we cannot accept this one. Next, try to equate the above equation to the next multiple of 7 which is 84, we have

Since the answer is greater than 10, then we cannot accept this one also. If you will continue this process to the next multiple of 7 like 77, 70, 63 and so on, all values of x are greater than 10. Therefore, the possible number is 1085 only. You can check this number by using a calculator and this number is divisible by 7.