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 ->  Human-and-algebraic-language -> 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      Log On


   

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) About Me  (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 4201012 is 2, which is an even number.


Since the last digit is one of the digits in the list above, this means that 4201012 is divisible by 2.


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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 4201012 to get 4%2B2%2B0%2B1%2B0%2B1%2B2=10.


Because the digits add up to 10, which is NOT divisible by 3, this means that 4201012 is NOT divisible by 3.


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To see if a number is divisible by 4, we just need to see if the last two digits are divisible by 4.


Since 12 is divisible by 4 (the last two digits of 4201012), this means that 4201012 is divisible by 4 also.


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A number is divisible by 5 if the last digit is either 0 or 5.


Since the last digit of 4201012 is 2, this means that 4201012 is NOT divisible by 5.


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A number is divisible by 6 if the number is divisible by BOTH 2 AND 3 (since 2*3=6).


The number 4201012 may be divisible by 2 (see above), but it is NOT divisible by 3 (see above). So 4201012 is NOT divisible by 6 (since BOTH need to be divisible).


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To see if a number is divisible by 8, we just need to see if the last three digits are divisible by 8.


Since 012 or 12 is NOT divisible by 8 (the last three digits of 4201012), this means that 4201012 is NOT divisible by 8 as well.


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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 4201012 to get 4%2B2%2B0%2B1%2B0%2B1%2B2=10.


Because the digits add up to 10, which is NOT divisible by 9, this means that 4201012 is NOT divisible by 9.


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A number is divisible by 10 if the last digit is a 0.


Since the last digit of 4201012 is NOT 0, this means that 4201012 is NOT divisible by 10.


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A number is divisible by 11 if the sum of the alternating digits equals the sum of the remaining alternating digits.
Since 4%2B0%2B0%2B2%3C%3E2%2B1%2B1, this means that 4201012 is NOT divisible by 11.


So the number 4201012 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 1001 is 1, which is an odd number.


Since the last digit is NOT in the list above, this means that 1001 is NOT divisible by 2.


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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 1001 to get 1%2B0%2B0%2B1=2.


Because the digits add up to 2, which is NOT divisible by 3, this means that 1001 is NOT divisible by 3.


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To see if a number is divisible by 4, we just need to see if the last two digits are divisible by 4.


Since 01 is NOT divisible by 4 (the last two digits of 1001), this means that 1001 is NOT divisible by 4 as well.


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A number is divisible by 5 if the last digit is either 0 or 5.


Since the last digit of 1001 is 1, this means that 1001 is NOT divisible by 5.


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A number is divisible by 6 if the number is divisible by BOTH 2 AND 3 (since 2*3=6).


The number 1001 is neither divisible by 2 nor 3 (see above). So 1001 is NOT divisible by 6.


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To see if a number is divisible by 8, we just need to see if the last three digits are divisible by 8.


Since 001 or 1 is NOT divisible by 8 (the last three digits of 1001), this means that 1001 is NOT divisible by 8 as well.


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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 1001 to get 1%2B0%2B0%2B1=2.


Because the digits add up to 2, which is NOT divisible by 9, this means that 1001 is NOT divisible by 9.


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A number is divisible by 10 if the last digit is a 0.


Since the last digit of 1001 is NOT 0, this means that 1001 is NOT divisible by 10.


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A number is divisible by 11 if the sum of the alternating digits equals the sum of the remaining alternating digits.
Since 1%2B0=0%2B1, this means that 1001 is divisible by 11.


So the number 1001 is only divisible by 11


Note: 1001 may be divisible by other prime numbers, but 11 is the only divisible factor in our list.

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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 10001 is 1, which is an odd number.


Since the last digit is NOT in the list above, this means that 10001 is NOT divisible by 2.


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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 10001 to get 1%2B0%2B0%2B0%2B1=2.


Because the digits add up to 2, which is NOT divisible by 3, this means that 10001 is NOT divisible by 3.


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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 01 is NOT divisible by 4 (the last two digits of 10001), this means that 10001 is NOT divisible by 4 as well.


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Divisibility check for 5:


A number is divisible by 5 if the last digit is either 0 or 5.


Since the last digit of 10001 is 1, this means that 10001 is NOT divisible by 5.


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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 10001 is neither divisible by 2 nor 3 (see above). So 10001 is NOT divisible by 6.


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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 001 is NOT divisible by 8 (the last three digits of 10001), this means that 10001 is NOT divisible by 8 as well.


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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 10001 to get 1%2B0%2B0%2B0%2B1=2.


Because the digits add up to 2, which is NOT divisible by 9, this means that 10001 is NOT divisible by 9.


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Divisibility check for 10:


A number is divisible by 10 if the last digit is a 0.


Since the last digit of 10001 is NOT 0, this means that 10001 is NOT divisible by 10.


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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 1%2B0%2B1%3C%3E0%2B0, this means that 10001 is NOT divisible by 11.


So the number 10001 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 10001.