# How to obtain three consecutive integers?

• Math100
In summary, the conversation discussed finding three consecutive integers that satisfy certain divisibility conditions. The Chinese Remainder Theorem was used to find the smallest possible triple with distinct numbers, which turned out to be (5, 3, 2).
Math100
Homework Statement
Obtain three consecutive integers, the first of which is divisible by a square, the second by a cube, and the third by a fourth power.
Relevant Equations
None.
Let ## a, a+1 ## and ## a+2 ## be the three consecutive integers.
Then
\begin{align*}
&5^{2}\mid a\implies a\equiv 0\pmod {25}\\
&3^{3}\mid (a+1)\implies a+1\equiv 0\pmod {27}\implies a\equiv 26\pmod {9}\\
&2^{4}\mid (a+2)\implies a+2\equiv 0\pmod {16}\implies a\equiv 14\pmod {16}.\\
\end{align*}
Applying the Chinese Remainder Theorem produces:
## n=25\cdot 27\cdot 16=10800 ##.
This means ## N_{1}=\frac{10800}{25}=432, N_{2}=\frac{10800}{27}=400 ## and ## N_{3}=\frac{10800}{16}=675 ##.
Now we have ## 432x_{1}\equiv 1\pmod {25}, 400x_{2}\equiv 1\pmod {27} ## and ## 675x_{3}\equiv 1\pmod {16} ##.
Observe that
\begin{align*}
&432x_{1}\equiv 1\pmod {25}\implies 7x_{1}\equiv 1\pmod {25}\\
&\implies 49x_{1}\equiv 7\pmod {25}\implies -x_{1}\equiv 7\pmod {25}\\
&\implies x_{1}\equiv 18\pmod {25},\\
&400x_{2}\equiv 1\pmod {27}\implies -5x_{2}\equiv 1\pmod {27}\\
&\implies -25x_{2}\equiv 5\pmod {27}\implies 2x_{2}\equiv 5\pmod {27}\\
&\implies 28x_{2}\equiv 70\pmod {27}\implies x_{2}\equiv 16\pmod {27},\\
&675x_{3}\equiv 1\pmod {16}\implies 3x_{3}\equiv 1\pmod {16}\\
&\implies 15x_{3}\equiv 5\pmod {16}\implies -x_{3}\equiv 5\pmod {16}\\
&\implies x_{3}\equiv 11\pmod {16}.\\
\end{align*}
Since ## x_{1}=18, x_{2}=16 ## and ## x_{3}=11 ##,
it follows that ## x\equiv (0+26\cdot 400\cdot 16+14\cdot 675\cdot 11)\pmod {10800}\equiv 270350\pmod {10800}\equiv 350\pmod {10800} ##.
Thus, ## a=350, a+1=351 ## and ## a+2=352 ##.
Therefore, the three consecutive integers are ## 350, 351 ## and ## 352 ##.

The answer is correct, but how did you come to choose ##(5,3,2)##? And there is a typo in the second line where you wrote ##\pmod 9## instead of ##\pmod{27}.##

Math100
fresh_42 said:
The answer is correct, but how did you come to choose ##(5,3,2)##? And there is a typo in the second line where you wrote ##\pmod 9## instead of ##\pmod{27}.##
I chose ## (5, 3, 2) ## from the book's answer, since it says ## 5^{2}\mid 350, 3^{3}\mid 351 ## and ## 2^{4}\mid 352 ##. But do you know why it should be ## (5, 3, 2) ##? Because honestly, I do not know why. It seems like because all ## 2, 3 ## and ## 5 ## are prime numbers.

Math100 said:
I chose ## (5, 3, 2) ## from the book's answer, since it says ## 5^{2}\mid 350, 3^{3}\mid 351 ## and ## 2^{4}\mid 352 ##. But do you know why it should be ## (5, 3, 2) ##? Because honestly, I do not know why. It seems like because all ## 2, 3 ## and ## 5 ## are prime numbers.
We are asked to find ##r^2\,|\,a\, , \,s^3\,|\,(a+1)\,|\,t^4\,|\,(a+2).##

I wonder why they all have to be pairwise distinct. I assume there is a reason, but I haven't looked into it. If so, then ##r,s,t## shouldn't be too far apart to obtain a solution that can be computed without computers. It makes sense to choose ##t## as small as possible in order to keep ##a## small. That gives us ##t^4=16\,|\,(a+2).## Then ##s=3## is the next small number to keep ##s^3## and ##a## possibly small. The only question left is then, should we choose ##r=4## or ##r=5?## But from ##r=4## we would get ##16|a## and ##16|(a+2)## which is impossible.

So ##(5,3,2)## is the smallest possible triple with distinct numbers. And it worked. There are probably more solutions. It is just the smallest one.

Math100

## 1. How do I find three consecutive integers?

To find three consecutive integers, you can use the formula n, n+1, n+2 where n is any integer. For example, if n = 5, then the consecutive integers would be 5, 6, 7.

## 2. What is the difference between consecutive integers?

The difference between consecutive integers is always 1. This means that if you have three consecutive integers, the difference between each integer will be 1.

## 3. Can negative integers be consecutive?

Yes, negative integers can be consecutive. For example, -3, -2, -1 are three consecutive integers.

## 4. How can I obtain three consecutive integers in a specific range?

To obtain three consecutive integers in a specific range, you can use the formula n, n+1, n+2 where n is any integer within the range. For example, if you want three consecutive integers between 10 and 20, you can use 10, 11, 12 or 19, 20, 21.

## 5. How many ways can I obtain three consecutive integers?

There are infinite ways to obtain three consecutive integers. This is because there are infinite possible values for n, which is the starting integer. However, if you have a specific range or criteria, the number of ways may be limited.

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