SOLUTION: Without using a calculator, test each of the following
numbers for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11:
a. 4,201,012 b. 1001
c. 10,001
Algebra.Com
Question 357527: Without using a calculator, test each of the following
numbers for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11:
a. 4,201,012 b. 1001
c. 10,001
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)
Divisibility check for 2:
A number is divisible by 2 if the last digit is 2, 4, 6, 8, or 0.
Notice how the last digit of is , which is an even number.
Since the last digit is one of the digits in the list above, this means that is divisible by 2.
---------------------------------
Divisibility check for 3:
A number is divisible by 3 if the sum of the digits is divisible by 3.
First add up the digits in to get .
Because the digits add up to , which is NOT divisible by 3, this means that is NOT divisible by 3.
---------------------------------
To see if a number is divisible by 4, we just need to see if the last two digits are divisible by 4.
Since is divisible by 4 (the last two digits of ), this means that is divisible by 4 also.
---------------------------------
A number is divisible by 5 if the last digit is either 0 or 5.
Since the last digit of is , this means that is NOT divisible by 5.
---------------------------------
A number is divisible by 6 if the number is divisible by BOTH 2 AND 3 (since 2*3=6).
The number may be divisible by 2 (see above), but it is NOT divisible by 3 (see above). So is NOT divisible by 6 (since BOTH need to be divisible).
---------------------------------
To see if a number is divisible by 8, we just need to see if the last three digits are divisible by 8.
Since or is NOT divisible by 8 (the last three digits of ), this means that is NOT divisible by 8 as well.
---------------------------------
Divisibility check for 9:
A number is divisible by 9 if the sum of the digits is divisible by 9.
First add up the digits in to get .
Because the digits add up to , which is NOT divisible by 9, this means that is NOT divisible by 9.
---------------------------------
A number is divisible by 10 if the last digit is a 0.
Since the last digit of is NOT 0, this means that is NOT divisible by 10.
---------------------------------
A number is divisible by 11 if the sum of the alternating digits equals the sum of the remaining alternating digits.
Since , this means that is NOT divisible by 11.
So the number is divisble by the following numbers: 2 and 4
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b)
Divisibility check for 2:
A number is divisible by 2 if the last digit is 2, 4, 6, 8, or 0.
Notice how the last digit of is , which is an odd number.
Since the last digit is NOT in the list above, this means that is NOT divisible by 2.
---------------------------------
Divisibility check for 3:
A number is divisible by 3 if the sum of the digits is divisible by 3.
First add up the digits in to get .
Because the digits add up to , which is NOT divisible by 3, this means that is NOT divisible by 3.
---------------------------------
To see if a number is divisible by 4, we just need to see if the last two digits are divisible by 4.
Since is NOT divisible by 4 (the last two digits of ), this means that is NOT divisible by 4 as well.
---------------------------------
A number is divisible by 5 if the last digit is either 0 or 5.
Since the last digit of is , this means that is NOT divisible by 5.
---------------------------------
A number is divisible by 6 if the number is divisible by BOTH 2 AND 3 (since 2*3=6).
The number is neither divisible by 2 nor 3 (see above). So is NOT divisible by 6.
---------------------------------
To see if a number is divisible by 8, we just need to see if the last three digits are divisible by 8.
Since or is NOT divisible by 8 (the last three digits of ), this means that is NOT divisible by 8 as well.
---------------------------------
Divisibility check for 9:
A number is divisible by 9 if the sum of the digits is divisible by 9.
First add up the digits in to get .
Because the digits add up to , which is NOT divisible by 9, this means that is NOT divisible by 9.
---------------------------------
A number is divisible by 10 if the last digit is a 0.
Since the last digit of is NOT 0, this means that is NOT divisible by 10.
---------------------------------
A number is divisible by 11 if the sum of the alternating digits equals the sum of the remaining alternating digits.
Since , this means that is divisible by 11.
So the number is only divisible by
Note: may be divisible by other prime numbers, but is the only divisible factor in our list.
==========================================================
c)
Divisibility check for 2:
A number is divisible by 2 if the last digit is 2, 4, 6, 8, or 0.
Notice how the last digit of is , which is an odd number.
Since the last digit is NOT in the list above, this means that is NOT divisible by 2.
---------------------------------
Divisibility check for 3:
A number is divisible by 3 if the sum of the digits is divisible by 3.
First add up the digits in to get .
Because the digits add up to , which is NOT divisible by 3, this means that is NOT divisible by 3.
---------------------------------
Divisibility check for 4:
To see if a number is divisible by 4, we just need to see if the last two digits are divisible by 4.
Since is NOT divisible by 4 (the last two digits of ), this means that is NOT divisible by 4 as well.
---------------------------------
Divisibility check for 5:
A number is divisible by 5 if the last digit is either 0 or 5.
Since the last digit of is , this means that is NOT divisible by 5.
---------------------------------
Divisibility check for 6:
A number is divisible by 6 if the number is divisible by BOTH 2 AND 3 (since 2*3=6).
The number is neither divisible by 2 nor 3 (see above). So is NOT divisible by 6.
---------------------------------
Divisibility check for 8:
To see if a number is divisible by 8, we just need to see if the last three digits are divisible by 8.
Since is NOT divisible by 8 (the last three digits of ), this means that is NOT divisible by 8 as well.
---------------------------------
Divisibility check for 9:
A number is divisible by 9 if the sum of the digits is divisible by 9.
First add up the digits in to get .
Because the digits add up to , which is NOT divisible by 9, this means that is NOT divisible by 9.
---------------------------------
Divisibility check for 10:
A number is divisible by 10 if the last digit is a 0.
Since the last digit of is NOT 0, this means that is NOT divisible by 10.
---------------------------------
Divisibility check for 11:
A number is divisible by 11 if the sum of the alternating digits equals the sum of the remaining alternating digits.
Since , this means that is NOT divisible by 11.
So the number is NOT divisible by 2, 3, 4, 5, 6, 8, 9, 10, or 11
In other words, none of the numbers on the list are factors of the number .
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