A number is divisible by 6 if it is divisible by 2 and 3.
Look at 333-3 which is 330 The sum of the digits is 6 and it is even so it is divisible by 6 Now consider 222-2 which I picked because unlike 333, 222 has even digits. 222-2=220, one again even number so divisible by 2 but NOT divisible by 3 so NOT divisible by 6
So it look like this is not true for all n
For any odd n, we have the following
1. nnn-n ends in 0 so it is even if we can show it is divisible by 3 we are done. but 777-7 is 770 which is NOT divisible by 3 so it is NOT true.
For some n it is true, but not for all n... Now when will nnn-n be divisible by 3.
only when n+n is a multiple of 3, ie n=33,66, 99 an that is it!
So we could easily prove that nnn-n is divisible by 6 if and only if n=3,6,or 9
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If by nnn, you mean n3, a proof is as follows:
n=0,1,2,3,4, or 5 (mod 6)
If n=0 (mod 6), we have (0 (mod 6))((0(mod 6))2-1)=0 (mod 6). [Since the first term is zero]
If n=1 (mod 6), we have (1 (mod 6))((1(mod 6))2-1)=0 (mod 6) [Since 1-1=0].
If n=2 (mod 6), we have (2 (mod 6))((2(mod 6))2-1)=(2*3) (mod 6) = 6 (mod 6)=0 (mod 6).
If n=3 (mod 6), we have (3 (mod 6))((3(mod 6))2-1)=(3*8) (mod 6) = 24 (mod 6) = 0 (mod 6).
If n=4 (mod 6), we have (4 (mod 6))((4(mod 6))2-1)=(4*15) (mod 6) = 60 (mod 6) = 0 (mod 6).
If n=5 (mod 6), we have (5 (mod 6))((5(mod 6))2-1)=(5*24) (mod 6) = 120 (mod 6) = 0 (mod 6).
If you're not comfortable with the modular arethmetic, you can substitue 6m+_, where the blank is each of the numbers 0 through 5 (since every number can be expressed either as a multiple of six, or as a multiple of six plus some number between 1 and 5 --the remainder when the number is divided by six). Taking our example with 5, you would get:
(n)(n2-1) can be written as (6m+5)((6m+5)2-1), where m is an integer.
Simplifying this, you get:
(6m+5)((6m+5)2-1)
(6m+5)((6m2+60m+25-1)
6m*6m2+6m*60m+6m*25-6m+5*6m2+5*60m+5*25+5(-1)
6m*6m2+6m*60m+6m*25-6m+5*6m2+5*60m+5*24
Since m is an integer and each term is divisible by 6, (n)(n2-1) is divisible by 6 for integers that can be expressed as 6m+5. You would then repeat the process for each of 0 through 4 to complete the proof. Clearly, if you are comfortable with it, modular arithmetic is the less cumbersome way to proceed.
help me
It's not. If n = 2, then 3n - 1 = 3*2 - 1 = 6 - 1 = 5, which isn't divisible by 2.
No. 483 is not divisible by 6.A number is divisible by 6 if it is divisible by both 2 and 3.It is divisible by 2 if it is even and it is divisible by 3 if the sum of the digits is a multiple of 3.483 is not divisible by 6 since it is not divisible by 2 although it is divisible by 3.
If a number is divisible by 2 and 3, it is divisible by 6.
Since 3 is a factor of 6, any number divisible by 6 is also divisible by 3. But since 2 is also a factor of 6, then any number divisible by 6 must also be divisible by 2. This means that any number divisible by 6 is an even number. So if a number is odd and it is divisible by 3, then it is not divisible by 6. For example, 12 is divisible by 3, but since it is even, it is also divisible by 6. But 15 is divisible by 3, and it is odd, so it is not divisible by 6.
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Because 6*8 = 48 and 48/8 = 6
It's not. If n = 2, then 3n - 1 = 3*2 - 1 = 6 - 1 = 5, which isn't divisible by 2.
6 is not divisible by 162. 162 is divisible by 6.
Any
No. 483 is not divisible by 6.A number is divisible by 6 if it is divisible by both 2 and 3.It is divisible by 2 if it is even and it is divisible by 3 if the sum of the digits is a multiple of 3.483 is not divisible by 6 since it is not divisible by 2 although it is divisible by 3.
If it is divisible by 2 and 3, it is divisible by 6.
if a number is divisible by 2 and 3 then its divisible by 6
If a number is divisible by 2 and 3, it is divisible by 6.
No odd number can be evenly divisible by 6. Since 6 is divisible by 2, any number that is divisible by 6 will automatically be divisible by 2.
Multiples of 9 and 6 are also divisible by three, the reverse is not true. 15 is divisible by 3, but not 6 or 9. 27 is divisible by 3 and 9, but not 6. 12 is divisible by 3 and 6, but not 9. 54 is divisible by 3, 6 and 9.
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