Given a three digit number $n$, let $f(n)$ be the sum of digits of $n$, their products in pairs, and the product of all digits. When does $n=f(n)$?
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This is my first time posting so do correct me if I am doing anything wrong.
Please help me with this math problem from the British Maths Olympiad (1994 British Maths Olympiad1 Q1 Number Theory).
Starting with any three digit number $n$ (such as $n = 625$) we obtain a new number $f(n)$ which is equal to the sum of the three digits of $n$, their three products in pairs, and the product of all three digits.
Find all three digit numbers such that $fracnf(n)=1$.
The only solution I found is $199$, can someone verify it please?
elementary-number-theory contest-math
New contributor
|
show 2 more comments
up vote
3
down vote
favorite
This is my first time posting so do correct me if I am doing anything wrong.
Please help me with this math problem from the British Maths Olympiad (1994 British Maths Olympiad1 Q1 Number Theory).
Starting with any three digit number $n$ (such as $n = 625$) we obtain a new number $f(n)$ which is equal to the sum of the three digits of $n$, their three products in pairs, and the product of all three digits.
Find all three digit numbers such that $fracnf(n)=1$.
The only solution I found is $199$, can someone verify it please?
elementary-number-theory contest-math
New contributor
in the definition of $f$, are you concatenating the three results? Can you give an example of a pair $(n,f(n))$?
– mathworker21
Nov 19 at 5:44
So $f(199) = 19, 9918, 81$?
– steven gregory
Nov 19 at 5:45
1
@mathworker21`@stevengregory For example if $n=625$, $f(n)=6+2+5+6*2+6*5+2*5+6*2*5$
– 3684
Nov 19 at 5:51
2
to do this problem, write $n = 100a+10b+c$ and just write everything out and solve the equation you get
– mathworker21
Nov 19 at 5:56
1
@mathworker21, I have tried that but I didn't get too far, can you try it if you have time? I also thought about factorising a+b+c+ab+ac+bc+abc as (a+1)(b+1)(c+1)-1 but still wasn't able to get far. For number theory Olympiad problems are there systematic methods or does it require a different method every time.
– 3684
Nov 19 at 6:00
|
show 2 more comments
up vote
3
down vote
favorite
up vote
3
down vote
favorite
This is my first time posting so do correct me if I am doing anything wrong.
Please help me with this math problem from the British Maths Olympiad (1994 British Maths Olympiad1 Q1 Number Theory).
Starting with any three digit number $n$ (such as $n = 625$) we obtain a new number $f(n)$ which is equal to the sum of the three digits of $n$, their three products in pairs, and the product of all three digits.
Find all three digit numbers such that $fracnf(n)=1$.
The only solution I found is $199$, can someone verify it please?
elementary-number-theory contest-math
New contributor
This is my first time posting so do correct me if I am doing anything wrong.
Please help me with this math problem from the British Maths Olympiad (1994 British Maths Olympiad1 Q1 Number Theory).
Starting with any three digit number $n$ (such as $n = 625$) we obtain a new number $f(n)$ which is equal to the sum of the three digits of $n$, their three products in pairs, and the product of all three digits.
Find all three digit numbers such that $fracnf(n)=1$.
The only solution I found is $199$, can someone verify it please?
elementary-number-theory contest-math
elementary-number-theory contest-math
New contributor
New contributor
edited Nov 19 at 12:31
amWhy
191k27223437
191k27223437
New contributor
asked Nov 19 at 5:36
3684
306
306
New contributor
New contributor
in the definition of $f$, are you concatenating the three results? Can you give an example of a pair $(n,f(n))$?
– mathworker21
Nov 19 at 5:44
So $f(199) = 19, 9918, 81$?
– steven gregory
Nov 19 at 5:45
1
@mathworker21`@stevengregory For example if $n=625$, $f(n)=6+2+5+6*2+6*5+2*5+6*2*5$
– 3684
Nov 19 at 5:51
2
to do this problem, write $n = 100a+10b+c$ and just write everything out and solve the equation you get
– mathworker21
Nov 19 at 5:56
1
@mathworker21, I have tried that but I didn't get too far, can you try it if you have time? I also thought about factorising a+b+c+ab+ac+bc+abc as (a+1)(b+1)(c+1)-1 but still wasn't able to get far. For number theory Olympiad problems are there systematic methods or does it require a different method every time.
– 3684
Nov 19 at 6:00
|
show 2 more comments
in the definition of $f$, are you concatenating the three results? Can you give an example of a pair $(n,f(n))$?
– mathworker21
Nov 19 at 5:44
So $f(199) = 19, 9918, 81$?
– steven gregory
Nov 19 at 5:45
1
@mathworker21`@stevengregory For example if $n=625$, $f(n)=6+2+5+6*2+6*5+2*5+6*2*5$
– 3684
Nov 19 at 5:51
2
to do this problem, write $n = 100a+10b+c$ and just write everything out and solve the equation you get
– mathworker21
Nov 19 at 5:56
1
@mathworker21, I have tried that but I didn't get too far, can you try it if you have time? I also thought about factorising a+b+c+ab+ac+bc+abc as (a+1)(b+1)(c+1)-1 but still wasn't able to get far. For number theory Olympiad problems are there systematic methods or does it require a different method every time.
– 3684
Nov 19 at 6:00
in the definition of $f$, are you concatenating the three results? Can you give an example of a pair $(n,f(n))$?
– mathworker21
Nov 19 at 5:44
in the definition of $f$, are you concatenating the three results? Can you give an example of a pair $(n,f(n))$?
– mathworker21
Nov 19 at 5:44
So $f(199) = 19, 9918, 81$?
– steven gregory
Nov 19 at 5:45
So $f(199) = 19, 9918, 81$?
– steven gregory
Nov 19 at 5:45
1
1
@mathworker21`@stevengregory For example if $n=625$, $f(n)=6+2+5+6*2+6*5+2*5+6*2*5$
– 3684
Nov 19 at 5:51
@mathworker21`@stevengregory For example if $n=625$, $f(n)=6+2+5+6*2+6*5+2*5+6*2*5$
– 3684
Nov 19 at 5:51
2
2
to do this problem, write $n = 100a+10b+c$ and just write everything out and solve the equation you get
– mathworker21
Nov 19 at 5:56
to do this problem, write $n = 100a+10b+c$ and just write everything out and solve the equation you get
– mathworker21
Nov 19 at 5:56
1
1
@mathworker21, I have tried that but I didn't get too far, can you try it if you have time? I also thought about factorising a+b+c+ab+ac+bc+abc as (a+1)(b+1)(c+1)-1 but still wasn't able to get far. For number theory Olympiad problems are there systematic methods or does it require a different method every time.
– 3684
Nov 19 at 6:00
@mathworker21, I have tried that but I didn't get too far, can you try it if you have time? I also thought about factorising a+b+c+ab+ac+bc+abc as (a+1)(b+1)(c+1)-1 but still wasn't able to get far. For number theory Olympiad problems are there systematic methods or does it require a different method every time.
– 3684
Nov 19 at 6:00
|
show 2 more comments
2 Answers
2
active
oldest
votes
up vote
9
down vote
accepted
Let $n=100a+10b+c,$ where $a> 0$ and $b,cgeq 0$. We are trying to solve $$100a+10b+c=a+b+c+ab+ac+bc+abc \ implies 99a+9b=abc+ab+ac+bc \ implies a(99-b-c-bc)=b(c-9) \$$$c-9leq 0$, but $b+c+bcleq 99$. So the above equation holds iff $b=c=9$, which means $a$ can take any value.
May I ask how you got to the solution so quick, do you just see the solution?
– 3684
Nov 19 at 6:13
@3684 this problem is fairly 'routine'. Plus, I might've seen this before/its associated solution, but I wouldn't remember if I did
– user574848
Nov 19 at 6:15
How many main 'routine' methods to solve number theory problems would you say there are? At higher levels do you think each question requires some different insight.
– 3684
Nov 19 at 6:19
add a comment |
up vote
8
down vote
Here's part $(b)$ because I'm assuming you don't need help with part $(a)$:
We want to compute all possible integers $n$ such that $fracnf(n) = 1$. Since we know that, by assumption, $n$ is a three-digit number, we can write
$$n = 100a + 10b + c,$$
where $a, b, c$ are integers. If this is the case, in terms of our newly defined variables $a$, $b$, and $c$, we can express $f(n)$ as follows:
$$f(n) = abc + ab + bc + ac + a + b + c.$$
Now, in order to have $fracnf(n) = 1,$ we must have $n = f(n)$. This happens when
$$99a + 9b = abc + ab + bc + ac$$
$$Longleftrightarrow (9-c)b = a(bc + b + c - 99) $$
Also, we must have $b, c leq 9,$ which implies $bc + b + c - 99 leq 0$. However, since $a neq 0$ (if $a = 0$, we would be able to form a two or one-digit number instead of a three-digit one!), we conclude $b = c = 9$. Therefore, our solution set is given by
$$boxed199, 299, 399, 499, 599, 699, 799, 899, 999$$
New contributor
Thank you for your answer, sorry I could only accept one answer. May I ask how you see the solution so quick?
– 3684
Nov 19 at 6:15
It's okay. Hopefully my answer still helps you. I also used to participate in math competitions. I think that the best way to get faster is to just practice by doing many problems.
– Ekesh
Nov 19 at 6:18
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
9
down vote
accepted
Let $n=100a+10b+c,$ where $a> 0$ and $b,cgeq 0$. We are trying to solve $$100a+10b+c=a+b+c+ab+ac+bc+abc \ implies 99a+9b=abc+ab+ac+bc \ implies a(99-b-c-bc)=b(c-9) \$$$c-9leq 0$, but $b+c+bcleq 99$. So the above equation holds iff $b=c=9$, which means $a$ can take any value.
May I ask how you got to the solution so quick, do you just see the solution?
– 3684
Nov 19 at 6:13
@3684 this problem is fairly 'routine'. Plus, I might've seen this before/its associated solution, but I wouldn't remember if I did
– user574848
Nov 19 at 6:15
How many main 'routine' methods to solve number theory problems would you say there are? At higher levels do you think each question requires some different insight.
– 3684
Nov 19 at 6:19
add a comment |
up vote
9
down vote
accepted
Let $n=100a+10b+c,$ where $a> 0$ and $b,cgeq 0$. We are trying to solve $$100a+10b+c=a+b+c+ab+ac+bc+abc \ implies 99a+9b=abc+ab+ac+bc \ implies a(99-b-c-bc)=b(c-9) \$$$c-9leq 0$, but $b+c+bcleq 99$. So the above equation holds iff $b=c=9$, which means $a$ can take any value.
May I ask how you got to the solution so quick, do you just see the solution?
– 3684
Nov 19 at 6:13
@3684 this problem is fairly 'routine'. Plus, I might've seen this before/its associated solution, but I wouldn't remember if I did
– user574848
Nov 19 at 6:15
How many main 'routine' methods to solve number theory problems would you say there are? At higher levels do you think each question requires some different insight.
– 3684
Nov 19 at 6:19
add a comment |
up vote
9
down vote
accepted
up vote
9
down vote
accepted
Let $n=100a+10b+c,$ where $a> 0$ and $b,cgeq 0$. We are trying to solve $$100a+10b+c=a+b+c+ab+ac+bc+abc \ implies 99a+9b=abc+ab+ac+bc \ implies a(99-b-c-bc)=b(c-9) \$$$c-9leq 0$, but $b+c+bcleq 99$. So the above equation holds iff $b=c=9$, which means $a$ can take any value.
Let $n=100a+10b+c,$ where $a> 0$ and $b,cgeq 0$. We are trying to solve $$100a+10b+c=a+b+c+ab+ac+bc+abc \ implies 99a+9b=abc+ab+ac+bc \ implies a(99-b-c-bc)=b(c-9) \$$$c-9leq 0$, but $b+c+bcleq 99$. So the above equation holds iff $b=c=9$, which means $a$ can take any value.
answered Nov 19 at 6:07
user574848
12712
12712
May I ask how you got to the solution so quick, do you just see the solution?
– 3684
Nov 19 at 6:13
@3684 this problem is fairly 'routine'. Plus, I might've seen this before/its associated solution, but I wouldn't remember if I did
– user574848
Nov 19 at 6:15
How many main 'routine' methods to solve number theory problems would you say there are? At higher levels do you think each question requires some different insight.
– 3684
Nov 19 at 6:19
add a comment |
May I ask how you got to the solution so quick, do you just see the solution?
– 3684
Nov 19 at 6:13
@3684 this problem is fairly 'routine'. Plus, I might've seen this before/its associated solution, but I wouldn't remember if I did
– user574848
Nov 19 at 6:15
How many main 'routine' methods to solve number theory problems would you say there are? At higher levels do you think each question requires some different insight.
– 3684
Nov 19 at 6:19
May I ask how you got to the solution so quick, do you just see the solution?
– 3684
Nov 19 at 6:13
May I ask how you got to the solution so quick, do you just see the solution?
– 3684
Nov 19 at 6:13
@3684 this problem is fairly 'routine'. Plus, I might've seen this before/its associated solution, but I wouldn't remember if I did
– user574848
Nov 19 at 6:15
@3684 this problem is fairly 'routine'. Plus, I might've seen this before/its associated solution, but I wouldn't remember if I did
– user574848
Nov 19 at 6:15
How many main 'routine' methods to solve number theory problems would you say there are? At higher levels do you think each question requires some different insight.
– 3684
Nov 19 at 6:19
How many main 'routine' methods to solve number theory problems would you say there are? At higher levels do you think each question requires some different insight.
– 3684
Nov 19 at 6:19
add a comment |
up vote
8
down vote
Here's part $(b)$ because I'm assuming you don't need help with part $(a)$:
We want to compute all possible integers $n$ such that $fracnf(n) = 1$. Since we know that, by assumption, $n$ is a three-digit number, we can write
$$n = 100a + 10b + c,$$
where $a, b, c$ are integers. If this is the case, in terms of our newly defined variables $a$, $b$, and $c$, we can express $f(n)$ as follows:
$$f(n) = abc + ab + bc + ac + a + b + c.$$
Now, in order to have $fracnf(n) = 1,$ we must have $n = f(n)$. This happens when
$$99a + 9b = abc + ab + bc + ac$$
$$Longleftrightarrow (9-c)b = a(bc + b + c - 99) $$
Also, we must have $b, c leq 9,$ which implies $bc + b + c - 99 leq 0$. However, since $a neq 0$ (if $a = 0$, we would be able to form a two or one-digit number instead of a three-digit one!), we conclude $b = c = 9$. Therefore, our solution set is given by
$$boxed199, 299, 399, 499, 599, 699, 799, 899, 999$$
New contributor
Thank you for your answer, sorry I could only accept one answer. May I ask how you see the solution so quick?
– 3684
Nov 19 at 6:15
It's okay. Hopefully my answer still helps you. I also used to participate in math competitions. I think that the best way to get faster is to just practice by doing many problems.
– Ekesh
Nov 19 at 6:18
add a comment |
up vote
8
down vote
Here's part $(b)$ because I'm assuming you don't need help with part $(a)$:
We want to compute all possible integers $n$ such that $fracnf(n) = 1$. Since we know that, by assumption, $n$ is a three-digit number, we can write
$$n = 100a + 10b + c,$$
where $a, b, c$ are integers. If this is the case, in terms of our newly defined variables $a$, $b$, and $c$, we can express $f(n)$ as follows:
$$f(n) = abc + ab + bc + ac + a + b + c.$$
Now, in order to have $fracnf(n) = 1,$ we must have $n = f(n)$. This happens when
$$99a + 9b = abc + ab + bc + ac$$
$$Longleftrightarrow (9-c)b = a(bc + b + c - 99) $$
Also, we must have $b, c leq 9,$ which implies $bc + b + c - 99 leq 0$. However, since $a neq 0$ (if $a = 0$, we would be able to form a two or one-digit number instead of a three-digit one!), we conclude $b = c = 9$. Therefore, our solution set is given by
$$boxed199, 299, 399, 499, 599, 699, 799, 899, 999$$
New contributor
Thank you for your answer, sorry I could only accept one answer. May I ask how you see the solution so quick?
– 3684
Nov 19 at 6:15
It's okay. Hopefully my answer still helps you. I also used to participate in math competitions. I think that the best way to get faster is to just practice by doing many problems.
– Ekesh
Nov 19 at 6:18
add a comment |
up vote
8
down vote
up vote
8
down vote
Here's part $(b)$ because I'm assuming you don't need help with part $(a)$:
We want to compute all possible integers $n$ such that $fracnf(n) = 1$. Since we know that, by assumption, $n$ is a three-digit number, we can write
$$n = 100a + 10b + c,$$
where $a, b, c$ are integers. If this is the case, in terms of our newly defined variables $a$, $b$, and $c$, we can express $f(n)$ as follows:
$$f(n) = abc + ab + bc + ac + a + b + c.$$
Now, in order to have $fracnf(n) = 1,$ we must have $n = f(n)$. This happens when
$$99a + 9b = abc + ab + bc + ac$$
$$Longleftrightarrow (9-c)b = a(bc + b + c - 99) $$
Also, we must have $b, c leq 9,$ which implies $bc + b + c - 99 leq 0$. However, since $a neq 0$ (if $a = 0$, we would be able to form a two or one-digit number instead of a three-digit one!), we conclude $b = c = 9$. Therefore, our solution set is given by
$$boxed199, 299, 399, 499, 599, 699, 799, 899, 999$$
New contributor
Here's part $(b)$ because I'm assuming you don't need help with part $(a)$:
We want to compute all possible integers $n$ such that $fracnf(n) = 1$. Since we know that, by assumption, $n$ is a three-digit number, we can write
$$n = 100a + 10b + c,$$
where $a, b, c$ are integers. If this is the case, in terms of our newly defined variables $a$, $b$, and $c$, we can express $f(n)$ as follows:
$$f(n) = abc + ab + bc + ac + a + b + c.$$
Now, in order to have $fracnf(n) = 1,$ we must have $n = f(n)$. This happens when
$$99a + 9b = abc + ab + bc + ac$$
$$Longleftrightarrow (9-c)b = a(bc + b + c - 99) $$
Also, we must have $b, c leq 9,$ which implies $bc + b + c - 99 leq 0$. However, since $a neq 0$ (if $a = 0$, we would be able to form a two or one-digit number instead of a three-digit one!), we conclude $b = c = 9$. Therefore, our solution set is given by
$$boxed199, 299, 399, 499, 599, 699, 799, 899, 999$$
New contributor
New contributor
answered Nov 19 at 6:08
Ekesh
3715
3715
New contributor
New contributor
Thank you for your answer, sorry I could only accept one answer. May I ask how you see the solution so quick?
– 3684
Nov 19 at 6:15
It's okay. Hopefully my answer still helps you. I also used to participate in math competitions. I think that the best way to get faster is to just practice by doing many problems.
– Ekesh
Nov 19 at 6:18
add a comment |
Thank you for your answer, sorry I could only accept one answer. May I ask how you see the solution so quick?
– 3684
Nov 19 at 6:15
It's okay. Hopefully my answer still helps you. I also used to participate in math competitions. I think that the best way to get faster is to just practice by doing many problems.
– Ekesh
Nov 19 at 6:18
Thank you for your answer, sorry I could only accept one answer. May I ask how you see the solution so quick?
– 3684
Nov 19 at 6:15
Thank you for your answer, sorry I could only accept one answer. May I ask how you see the solution so quick?
– 3684
Nov 19 at 6:15
It's okay. Hopefully my answer still helps you. I also used to participate in math competitions. I think that the best way to get faster is to just practice by doing many problems.
– Ekesh
Nov 19 at 6:18
It's okay. Hopefully my answer still helps you. I also used to participate in math competitions. I think that the best way to get faster is to just practice by doing many problems.
– Ekesh
Nov 19 at 6:18
add a comment |
3684 is a new contributor. Be nice, and check out our Code of Conduct.
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in the definition of $f$, are you concatenating the three results? Can you give an example of a pair $(n,f(n))$?
– mathworker21
Nov 19 at 5:44
So $f(199) = 19, 9918, 81$?
– steven gregory
Nov 19 at 5:45
1
@mathworker21`@stevengregory For example if $n=625$, $f(n)=6+2+5+6*2+6*5+2*5+6*2*5$
– 3684
Nov 19 at 5:51
2
to do this problem, write $n = 100a+10b+c$ and just write everything out and solve the equation you get
– mathworker21
Nov 19 at 5:56
1
@mathworker21, I have tried that but I didn't get too far, can you try it if you have time? I also thought about factorising a+b+c+ab+ac+bc+abc as (a+1)(b+1)(c+1)-1 but still wasn't able to get far. For number theory Olympiad problems are there systematic methods or does it require a different method every time.
– 3684
Nov 19 at 6:00