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.
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.
Category: Arithmetic
"Published in Vacaville, California, USA"
Find the missing digit so that it becomes divisible by 11 for
a. 23?329
b. 39085?
Solution:
a. Consider the given number
To test the divisibility of a number by 11, group the sum of the alternating digits into two groups and then get their difference. If the result is a multiple of 11, then the given number is divisible by 11. Let's do this for the given number as follows
Since -11 is a multiple of 11, then we don't have to add any number so that it becomes a multiple of 11 and hence, 0 is the missing digit. Therefore, the possible number is 230329.
b. Consider the given number
To
test the divisibility of a number by 11, group the sum of the
alternating digits into two groups and then get their difference. If the
result is a multiple of 11, then the given number is divisible by 11.
Let's do this for the given number as follows
Since -9 is not a multiple of 11, then we need to add a number so that it becomes a multiple of 11. If you add -2 (add 2 at the second group), then the answer is -11. 2 is the highest digit that we can use. Therefore, the possible number is 390852.
Category: Arithmetic
"Published in Vacaville, California, USA"
Find the missing digit so that it becomes divisible by 10 for
a. 45?897
b. 32681?
Solution:
a. Consider the given number
Since
the last digit of the given number is 7, then it is not divisible by 10.
A number is divisible by 10 if the last digit is 0. There's
nothing that we can do in
order to become divisible by 10 since the last digit of a given number is
not 0. You can assign any number to the missing digit but
still, the given number will never become divisible by 10.
b. Consider the given number
A
number is divisible by 10 if the last digit is 0. Since the missing
digit is the last digit, then we can assign 0 so that the given
number becomes divisible by 10. Therefore, the possible number is 326810 only.
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.
