# Find the smallest integer ## a>2 ## such that ## 2\mid a ##

• Math100
The rest is self-evident.But yeah, it's not wrong. I just thought that it could be slightly improved.
Math100
Homework Statement
Find the smallest integer ## a>2 ## such that ## 2\mid a, 3\mid (a+1), 4\mid (a+2), 5\mid (a+3), 6\mid (a+4) ##.
Relevant Equations
None.
Let ## a>2 ## be the smallest integer.
Then
\begin{align*}
&2\mid a\implies a\equiv 0\pmod {2}\implies a\equiv 2\pmod {2}\\
&3\mid (a+1)\implies a+1\equiv 0\pmod {3}\implies a\equiv -1\pmod {3}\implies a\equiv 2\pmod {3}\\
&4\mid (a+2)\implies a+2\equiv 0\pmod {4}\implies a\equiv -2\pmod {4}\implies a\equiv 2\pmod {4}\\
&5\mid (a+3)\implies a+3\equiv 0\pmod {5}\implies a\equiv -3\pmod {5}\implies a\equiv 2\pmod {5}\\
&6\mid (a+4)\implies a+4\equiv 0\pmod {6}\implies a\equiv -4\pmod {6}\implies a\equiv 2\pmod {6}.\\
\end{align*}
Observe that ## lcm(2, 3, 4, 5, 6)=60 ##.
Thus ## a\equiv 2\pmod {60}\implies a=62 ##.
Therefore, the smallest integer ## a>2 ## such that ## 2\mid a, 3\mid (a+1), 4\mid (a+2), 5\mid (a+3), 6\mid (a+4) ## is ## 62 ##.

Math100 said:
Homework Statement:: Find the smallest integer ## a>2 ## such that ## 2\mid a, 3\mid (a+1), 4\mid (a+2), 5\mid (a+3), 6\mid (a+4) ##.
Relevant Equations:: None.

Let ## a>2 ## be the smallest integer.
Then
\begin{align*}
&2\mid a\implies a\equiv 0\pmod {2}\implies a\equiv 2\pmod {2}\\
&3\mid (a+1)\implies a+1\equiv 0\pmod {3}\implies a\equiv -1\pmod {3}\implies a\equiv 2\pmod {3}\\
&4\mid (a+2)\implies a+2\equiv 0\pmod {4}\implies a\equiv -2\pmod {4}\implies a\equiv 2\pmod {4}\\
&5\mid (a+3)\implies a+3\equiv 0\pmod {5}\implies a\equiv -3\pmod {5}\implies a\equiv 2\pmod {5}\\
&6\mid (a+4)\implies a+4\equiv 0\pmod {6}\implies a\equiv -4\pmod {6}\implies a\equiv 2\pmod {6}.\\
\end{align*}
Observe that ## lcm(2, 3, 4, 5, 6)=60 ##.
Thus ## a\equiv 2\pmod {60}\implies a=62 ##.
Therefore, the smallest integer ## a>2 ## such that ## 2\mid a, 3\mid (a+1), 4\mid (a+2), 5\mid (a+3), 6\mid (a+4) ## is ## 62 ##.
Nice. And correct.

You could drop the first condition ##a\equiv 0\pmod 2## since it follows automatically from ##4\,|\,(a+2).## Same for ##a\equiv 2\pmod 3.## We only need ##4,5,6.##

(##6\,|\,(a+4)\Longrightarrow 3\,|\,(a+3+1)\Longrightarrow 3\,|\,(a+1)##)

Last edited:
Math100
fresh_42 said:
You could drop the first condition ##a\equiv 0\pmod 2## since it follows automatically from ##4\,|\,(a+2).## Same for ##a\equiv 2\pmod 3.## We only need ##4,5,6.##

I'm sorry but what do you mean by it follows automatically from 4 | a=(a+2) for the first condition? Could you show how you derived that?

Nanitf said:
I'm sorry but what do you mean by it follows automatically from 4 | a=(a+2) for the first condition? Could you show how you derived that?
Assume ##a\not\equiv 0 \pmod{2}## then ##a=2k+1## for some ##k\in \mathbb{Z}## and ##a+2=2k+3## is odd, and thus cannot be divided by ##4.## Or positively:
\begin{align*}
4\,|\,(a+2) \Longrightarrow 2\,|\,(a+2) \Longrightarrow 2\,|\,a \Longrightarrow a\equiv 0\pmod{2}
\end{align*}
where the only interesting step is ##2\,|\,(a+2) \Rightarrow a+2=2\cdot b\Rightarrow a=2\cdot b-2=2\cdot(b-1) \Rightarrow 2\,|\,a.##

## 1. What does it mean for a number to be divisible by 2?

A number is divisible by 2 if it can be divided evenly by 2 without leaving a remainder. In other words, if the number divided by 2 results in a whole number, then it is divisible by 2.

## 2. Why does the question specify that the integer must be greater than 2?

The question specifies that the integer must be greater than 2 because 2 is the smallest even number. Any number less than 2 would not be divisible by 2, as it would not be an even number.

## 3. How do you find the smallest integer greater than 2 that is divisible by 2?

To find the smallest integer greater than 2 that is divisible by 2, you simply need to find the next even number after 2. In this case, the smallest integer greater than 2 that is divisible by 2 is 4.

## 4. Can you explain why 3 is not the smallest integer greater than 2 that is divisible by 2?

3 is not divisible by 2 because it is an odd number. In order for a number to be divisible by 2, it must be an even number. Since 3 is not even, it cannot be the smallest integer greater than 2 that is divisible by 2.

## 5. Is there a mathematical formula or rule to determine the smallest integer greater than 2 that is divisible by 2?

Yes, the mathematical rule is that any even number can be represented as 2n, where n is an integer. Therefore, to find the smallest integer greater than 2 that is divisible by 2, you simply need to find the next even number after 2, which is 4.

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