# Find Integer $n$: Reversed Digits = 999 - Prime < 6000

• MHB
• lfdahl
In summary, the conversation involved determining an integer $n$ with specific properties, such as being a prime less than $6000$ and having the last two digits of $n$ be less than $10$. It was also mentioned that if the decimal digits of $n$ are reversed to obtain $N$, then $N-n=999$. The solution was found to be a four-digit number with a last digit of 3. The participants thanked each other for their contributions and solutions.
lfdahl
Gold Member
MHB
Determine the integer $n$ with the properties:

a). $n$ is a prime less than $6000$,
b). the number formed by the last two digits of $n$ is $< 10$, and
c). if the decimal digits of $n$ are reversed to obtain $N$, then $N − n = 999$.

2003

greg1313 said:
2003

Thankyou very much, greg1313, for a correct answer!(Yes)

Would you perhaps like to show, how you came up with $2003$? (Talking)

lfdahl said:
Determine the integer $n$ with the properties:

a). $n$ is a prime less than $6000$,
b). the number formed by the last two digits of $n$ is $< 10$, and
c). if the decimal digits of $n$ are reversed to obtain $N$, then $N − n = 999$.
Let $n=\overline{abcd}$.
Then due to (a) we have a=0-5 and d=1,3,7,9.
And due to (b) we have c=0.
From (c) we get that $\overline{ab0d} + 999 = \overline{d0ba}$, so that:
- $d+9\to a$, implying that $d=a+1$, and thus $a=0,2$.
- $0+9+1\to b$, implying $b=0$.

0001 is not a prime and doesn't actually satisfy (c), so that only leaves 2003, which is the answer.

lfdahl said:
Thankyou very much, greg1313, for a correct answer!(Yes)
Would you perhaps like to show, how you came up with $2003$? (Talking)

[sp]From c), n is four digits, From a), the last digit of n is 3. The solution follows.[/sp]

I like Serena said:
Let $n=\overline{abcd}$.
Then due to (a) we have a=0-5 and d=1,3,7,9.
And due to (b) we have c=0.
From (c) we get that $\overline{ab0d} + 999 = \overline{d0ba}$, so that:
- $d+9\to a$, implying that $d=a+1$, and thus $a=0,2$.
- $0+9+1\to b$, implying $b=0$.

0001 is not a prime and doesn't actually satisfy (c), so that only leaves 2003, which is the answer.

Thankyou, I like Serena! - for your participation and nice solution!

P.S.:

- isn´t condition (c) satisfied by $0001$? $1000 - 0001 = 999$?

lfdahl said:
Thankyou, I like Serena! - for your participation and nice solution!

P.S.:

- isn´t condition (c) satisfied by $0001$? $1000 - 0001 = 999$?

lfdahl said:
- isn´t condition (c) satisfied by $0001$? $1000 - 0001 = 999$?
Isn't the number actually $1$ instead of $0001$?
Then, if we reverse its digits and subtract, don't we get $1-1=0\ne 999$?

After all, if we'd reverse and subtract $02003$, we'd get $30020 - 02003 = 28017 \ne 999$.
So it seems to me that it can't be the intention to have leading zeroes, can it?

I like Serena said:
Isn't the number actually $1$ instead of $0001$?
Then, if we reverse its digits and subtract, don't we get $1-1=0\ne 999$?

After all, if we'd reverse and subtract $02003$, we'd get $30020 - 02003 = 28017 \ne 999$.
So it seems to me that it can't be the intention to have leading zeroes, can it?
Thanks, I like Serena, I see your point.(Yes)

## 1. What is the goal of this research?

The goal of this research is to find an integer, n, such that when its digits are reversed, the resulting number is equal to 999 minus a prime number less than 6000.

## 2. How is the integer n related to the prime number in this equation?

The integer n is the number that, when its digits are reversed, is equal to 999 minus the prime number. This means that the sum of n and the prime number will be equal to 999.

## 3. Why is the prime number limited to being less than 6000?

The prime number is limited to being less than 6000 because it ensures that the resulting number, when subtracted from 999, will be a three-digit number. This makes it easier to work with and compare to the reversed digits of n.

## 4. Is there a specific method used to find the integer n?

Yes, there are several methods that can be used to find the integer n. Some possible approaches include trial and error, using mathematical equations and algorithms, or using computer programming and coding.

## 5. What are the potential applications of this research?

This research could have various applications, such as in cryptography, number theory, or even in creating puzzles or brain teasers. It could also potentially lead to further discoveries and advancements in mathematics.

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