Prove that $N - M$ $equiv$ $0$ mod $9$.
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We say that a number $N$ is a "rainbow" number if every digit from $0, 1, .... 9$ shows up exactly once in $N$; for example $1234567890$ and $1029384756$ are both rainbow numbers. Take any two integer rainbow numbers $N$, $M$. Prove that $N - M$ $equiv$ $0$ mod $9$.
I am trying to use direct prove to find an example of $N$ and $M$ in order to prove this question. Are there any two rainbow numbers that can satisfy above condition? I am kinda stuck here.
number-theory
edited Aug 14 at 3:17
Shaun
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asked Aug 14 at 3:02
yoman
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up vote
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We say that a number $N$ is a "rainbow" number if every digit from $0, 1, .... 9$ shows up exactly once in $N$; for example $1234567890$ and $1029384756$ are both rainbow numbers. Take any two integer rainbow numbers $N$, $M$. Prove that $N - M$ $equiv$ $0$ mod $9$.
I am trying to use direct prove to find an example of $N$ and $M$ in order to prove this question. Are there any two rainbow numbers that can satisfy above condition? I am kinda stuck here.
number-theory
edited Aug 14 at 3:17
Shaun
7,41092972
asked Aug 14 at 3:02
yoman
1
1
Hint: show that any rainbow number is divisble by 9.
– Sean Roberson
Aug 14 at 3:08
add a comment |Â
up vote
-3
down vote
favorite
up vote
-3
down vote
favorite
We say that a number $N$ is a "rainbow" number if every digit from $0, 1, .... 9$ shows up exactly once in $N$; for example $1234567890$ and $1029384756$ are both rainbow numbers. Take any two integer rainbow numbers $N$, $M$. Prove that $N - M$ $equiv$ $0$ mod $9$.
I am trying to use direct prove to find an example of $N$ and $M$ in order to prove this question. Are there any two rainbow numbers that can satisfy above condition? I am kinda stuck here.
number-theory
edited Aug 14 at 3:17
Shaun
7,41092972
asked Aug 14 at 3:02
yoman
1
We say that a number $N$ is a "rainbow" number if every digit from $0, 1, .... 9$ shows up exactly once in $N$; for example $1234567890$ and $1029384756$ are both rainbow numbers. Take any two integer rainbow numbers $N$, $M$. Prove that $N - M$ $equiv$ $0$ mod $9$.
I am trying to use direct prove to find an example of $N$ and $M$ in order to prove this question. Are there any two rainbow numbers that can satisfy above condition? I am kinda stuck here.
number-theory
edited Aug 14 at 3:17
Shaun
7,41092972
asked Aug 14 at 3:02
yoman
1
edited Aug 14 at 3:17
Shaun
7,41092972
edited Aug 14 at 3:17
Shaun
7,41092972
edited Aug 14 at 3:17
Shaun
7,41092972
7,41092972
asked Aug 14 at 3:02
yoman
1
asked Aug 14 at 3:02
yoman
1
asked Aug 14 at 3:02
yoman
1
1
1
Hint: show that any rainbow number is divisble by 9.
– Sean Roberson
Aug 14 at 3:08
add a comment |Â
1
Hint: show that any rainbow number is divisble by 9.
– Sean Roberson
Aug 14 at 3:08
1
1
Hint: show that any rainbow number is divisble by 9.
– Sean Roberson
Aug 14 at 3:08
Hint: show that any rainbow number is divisble by 9.
– Sean Roberson
Aug 14 at 3:08
add a comment |Â
1 Answer
1
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up vote
3
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You can't prove it by looking at particular two examples. You have to prove things in general or enumerate things (not a wise move). You can however try to construct some examples (and you should) to help you understand things.
If you want to prove a general statement is false, you can find a counter example, but this is not the case here.
Guide:
- A number is divisible by $9$ if the sum of each digit is divisible by $9$.
- Subtraction of two number that is divisible by $m$ is divisible by $m$.
answered Aug 14 at 3:07
Siong Thye Goh
79k134997
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
You can't prove it by looking at particular two examples. You have to prove things in general or enumerate things (not a wise move). You can however try to construct some examples (and you should) to help you understand things.
If you want to prove a general statement is false, you can find a counter example, but this is not the case here.
Guide:
- A number is divisible by $9$ if the sum of each digit is divisible by $9$.
- Subtraction of two number that is divisible by $m$ is divisible by $m$.
answered Aug 14 at 3:07
Siong Thye Goh
79k134997
add a comment |Â
up vote
3
down vote
You can't prove it by looking at particular two examples. You have to prove things in general or enumerate things (not a wise move). You can however try to construct some examples (and you should) to help you understand things.
If you want to prove a general statement is false, you can find a counter example, but this is not the case here.
Guide:
- A number is divisible by $9$ if the sum of each digit is divisible by $9$.
- Subtraction of two number that is divisible by $m$ is divisible by $m$.
answered Aug 14 at 3:07
Siong Thye Goh
79k134997
add a comment |Â
up vote
3
down vote
up vote
3
down vote
You can't prove it by looking at particular two examples. You have to prove things in general or enumerate things (not a wise move). You can however try to construct some examples (and you should) to help you understand things.
If you want to prove a general statement is false, you can find a counter example, but this is not the case here.
Guide:
- A number is divisible by $9$ if the sum of each digit is divisible by $9$.
- Subtraction of two number that is divisible by $m$ is divisible by $m$.
answered Aug 14 at 3:07
Siong Thye Goh
79k134997
You can't prove it by looking at particular two examples. You have to prove things in general or enumerate things (not a wise move). You can however try to construct some examples (and you should) to help you understand things.
If you want to prove a general statement is false, you can find a counter example, but this is not the case here.
Guide:
- A number is divisible by $9$ if the sum of each digit is divisible by $9$.
- Subtraction of two number that is divisible by $m$ is divisible by $m$.
answered Aug 14 at 3:07
Siong Thye Goh
79k134997
edited Aug 14 at 3:13
answered Aug 14 at 3:07
Siong Thye Goh
79k134997
answered Aug 14 at 3:07
Siong Thye Goh
79k134997
answered Aug 14 at 3:07
Siong Thye Goh
79k134997
79k134997
add a comment |Â
add a comment |Â
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1
Hint: show that any rainbow number is divisble by 9.
– Sean Roberson
Aug 14 at 3:08