__Category__: Arithmetic

"Published in Vacaville, California, USA"

Find the missing digit so that it becomes divisible by 9 for
a. 13__?__84
b. 2096__?__
__Solution__:
a. Consider the given number

A
number is divisible by 9 if the sum of the digits is a multiple of 9.
If you add the rest of the digits, the sum will be equal to
Since
16 is not a multiple of 9, then we need to add a number so that it
becomes a multiple of 9. So, 16 + 2 = 18. 2
is the highest digit that we can use because 2 + 9 = 11 will be a two
digit number and we need to use only one digit to fill up the missing
digit. Therefore, the possible number is **13284 **only.

b. Consider the given number

A
number is divisible by 9 if the sum of the digits is a multiple of 9.
If you add the rest of the digits, the sum will be equal to

Since
17 is not a multiple of 9, then we need to add a number so that it
becomes a multiple of 9. So, 17 + 1 = 18. 1
is the highest digit that we can use because 1 + 9 = 10 will be a two
digit number and we need to use only one digit to fill up the missing
digit. Therefore, the possible number is **20961 **only.

__Category__: Arithmetic

"Published in Newark, California, USA"

Find the missing digit so that it becomes divisible by 8 for
a. 78__?__45
b. 2468__?__
__Solution__:
a. Consider the given number
Since
the last digit of a given number is not an even number, then the
given number is not divisible by 8. The multiples of 8 are all even number. There's nothing that we can do in
order to become divisible by 8 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 8.
b. Consider the given number
A number is divisible by 8 if the last three digit is a multiple of 8. Since the missing digit is the last digit, then we can assign even number digits so that the last three digit will be divisible by 8. If the last digit is 0, then 68 becomes 680 and 680 is divisible by 8. If the last digit is 2, then 68 becomes 682 and 682 is not divisible by 8. If the last digit is 4, then 68 becomes 684 and 684 is not divisible by 8. If the last digit is 6, then 68 becomes 686 and 686 is not divisible by 8. If the last digit is 8, then 68 becomes 688 and 688 is divisible by 8. Therefore, the possible numbers are **24680** and **24688**.

__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.
__Category__: Arithmetic

"Published in Newark, California, USA"

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**.

__Category__: Arithmetic

"Published in Newark, California, USA"

Find the missing digit so that it becomes divisible by 5 for
a. 94__?__763
b. 32178__?__
__Solution__:
a. Consider the given number
Since the last digit of the given number is 3, then it is not divisible by 5. A number is divisible by 5 if the last digit is 5 or 0. There's
nothing that we can do in
order to become divisible by 5 since the last digit of a given number is
not 5 or 0. You can assign any number to the missing digit but
still, the given number will never become divisible by 5.
b. Consider the given number
A number is divisible by 5 if the last digit is 5 or 0. Since the missing digit is the last digit, then we can assign 5 and 0 so that the given number becomes divisible by 5. Therefore, the possible numbers are **321785** and **321780**.
__Category__: Arithmetic

"Published in Newark, California, USA"
Find the missing digit so that it becomes divisible by 4 for
a. 5__?__627
b. 6721__?__
__Solution__:
a. Consider the given number
Since
the last two digit of a given number which is 27 is not a multiple of 4, then the
given number is not divisible by 4. There's nothing that we can do in
order to become divisible by 4 since the last digit of a given number is not a multiple of 4. You can assign any number to the missing digit but
still, the given number will never become divisible by 4.
b. Consider the given number
A number is divisible by 4 if the last two digit is a multiple of 4. Since 1 is located at the second to the last digit, then we can assign 2 and 6 to the last digit so that 1 becomes 12 and 16 which are the multiples of 4. Therefore, the possible numbers are **67212** and **67216**.

__Category__: Arithmetic

"Published in Newark, California, USA"
Find the missing digit so that it becomes divisible by 3 for
a. 35__?__83
b. 7895__?__
__Solution__:
a. Consider the given number
A number is divisible by 3 if the sum of the digits is a multiple of 3. If you add the rest of the digits, the sum will be equal to
Since 19 is not a multiple of 3, then we need to add a number so that it becomes a multiple of 3. So, 19 + 2 = 21. We can add also 5 (2 + 3) so that 19 + 5 = 24. We can add also 8 (2 + 3 + 3) so that 19 + 8 = 27. 8 is the highest digit that we can use because 8 + 3 = 11 will be a two digit number and we need to use only one digit to fill up the missing digit. Therefore, the possible numbers are **35283**, **35583**, and **35883**.
b. Consider the given number
A
number is divisible by 3 if the sum of the digits is a multiple of 3.
If you add the rest of the digits, the sum will be equal to
Since 29 is not a multiple of 3, then we need to add a number so that it becomes a multiple of 3. So, 29 + 1 = 30. We can add also 4 (1 + 3) so that 29 + 4 = 33. We can add also 7 (1 + 3 + 3) so that 29 + 7 = 36. 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. Therefore, the possible numbers are **78951**, **78954**, and **78957**.