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Is $n^2+3n+6$ divisible by 25, where $n$ is a integer?
$n^2 + 3n +5$ is not divisible by $121$Counting the amount of N digit numbers made from K digits divisible by X.Proof of $pi$+$e$ irrationalProve that $sqrt2 + 9n$ is never an integerHow to get a number that is divisible by $n$ - without obviously seeing it?Prove that if $n geq 2$, then $sqrt[n]n$ is irrational. Hint, show that if $n geq 2$, then $2^n > n$.F inding out possible remainders when dividing an integerFastest way to count divisors of number that are not divisible by one prime number.Finding $n$ such that atleast one permutation of the digits $1,2,ldots,n$ is divisible by $11$.Trying to prove that if $n^2=3q$, then $n=3p$ for $n,p,q inmathbbN$.Finding the last 4 digits of a huge power
$begingroup$
If we want $n^2+3n+6$ to be divisible by $25$, it firstly has to be divisible by $5$.
So, let's take a look at couple of cases:
$n=5k: 25k^2+15k+6$. The remainder is $6$, so it's not divisible by $25$.
$n=5k+1: 25k^2+25k+10$. The remainder is $10$, so it is not divisible by $25$.
And so on for $n=5k+2, n=5k+3, n=5k+4.$
- Is this a good way to do this?
The other way that came to my mind would be following:
Let's say that $n^2+3n+6$ is divisible by $25$: $$n^2+3n+6=25k$$
$$n^2+3n+(6-25k)=0$$
If we solve this equation, we get $$n_1,2=frac-3 pm sqrt5 sqrt20k-32.$$ But $sqrt5$ is a irrational number, so is $n$ irrational number too. This is contradiction, so it's not divisible by $25$.
- Is this a good way to solve this problem?
number-theory elementary-number-theory divisibility
$endgroup$
add a comment |
$begingroup$
If we want $n^2+3n+6$ to be divisible by $25$, it firstly has to be divisible by $5$.
So, let's take a look at couple of cases:
$n=5k: 25k^2+15k+6$. The remainder is $6$, so it's not divisible by $25$.
$n=5k+1: 25k^2+25k+10$. The remainder is $10$, so it is not divisible by $25$.
And so on for $n=5k+2, n=5k+3, n=5k+4.$
- Is this a good way to do this?
The other way that came to my mind would be following:
Let's say that $n^2+3n+6$ is divisible by $25$: $$n^2+3n+6=25k$$
$$n^2+3n+(6-25k)=0$$
If we solve this equation, we get $$n_1,2=frac-3 pm sqrt5 sqrt20k-32.$$ But $sqrt5$ is a irrational number, so is $n$ irrational number too. This is contradiction, so it's not divisible by $25$.
- Is this a good way to solve this problem?
number-theory elementary-number-theory divisibility
$endgroup$
3
$begingroup$
Your second method needs to consider $sqrt20k-3$ too as that might balance the $sqrt5$ and make the overall result rational - in fact it does not, but you have not shown it does not
$endgroup$
– Henry
Mar 11 at 9:58
add a comment |
$begingroup$
If we want $n^2+3n+6$ to be divisible by $25$, it firstly has to be divisible by $5$.
So, let's take a look at couple of cases:
$n=5k: 25k^2+15k+6$. The remainder is $6$, so it's not divisible by $25$.
$n=5k+1: 25k^2+25k+10$. The remainder is $10$, so it is not divisible by $25$.
And so on for $n=5k+2, n=5k+3, n=5k+4.$
- Is this a good way to do this?
The other way that came to my mind would be following:
Let's say that $n^2+3n+6$ is divisible by $25$: $$n^2+3n+6=25k$$
$$n^2+3n+(6-25k)=0$$
If we solve this equation, we get $$n_1,2=frac-3 pm sqrt5 sqrt20k-32.$$ But $sqrt5$ is a irrational number, so is $n$ irrational number too. This is contradiction, so it's not divisible by $25$.
- Is this a good way to solve this problem?
number-theory elementary-number-theory divisibility
$endgroup$
If we want $n^2+3n+6$ to be divisible by $25$, it firstly has to be divisible by $5$.
So, let's take a look at couple of cases:
$n=5k: 25k^2+15k+6$. The remainder is $6$, so it's not divisible by $25$.
$n=5k+1: 25k^2+25k+10$. The remainder is $10$, so it is not divisible by $25$.
And so on for $n=5k+2, n=5k+3, n=5k+4.$
- Is this a good way to do this?
The other way that came to my mind would be following:
Let's say that $n^2+3n+6$ is divisible by $25$: $$n^2+3n+6=25k$$
$$n^2+3n+(6-25k)=0$$
If we solve this equation, we get $$n_1,2=frac-3 pm sqrt5 sqrt20k-32.$$ But $sqrt5$ is a irrational number, so is $n$ irrational number too. This is contradiction, so it's not divisible by $25$.
- Is this a good way to solve this problem?
number-theory elementary-number-theory divisibility
number-theory elementary-number-theory divisibility
edited Mar 11 at 10:21
Gurjinder
554417
554417
asked Mar 11 at 9:53
user389231
3
$begingroup$
Your second method needs to consider $sqrt20k-3$ too as that might balance the $sqrt5$ and make the overall result rational - in fact it does not, but you have not shown it does not
$endgroup$
– Henry
Mar 11 at 9:58
add a comment |
3
$begingroup$
Your second method needs to consider $sqrt20k-3$ too as that might balance the $sqrt5$ and make the overall result rational - in fact it does not, but you have not shown it does not
$endgroup$
– Henry
Mar 11 at 9:58
3
3
$begingroup$
Your second method needs to consider $sqrt20k-3$ too as that might balance the $sqrt5$ and make the overall result rational - in fact it does not, but you have not shown it does not
$endgroup$
– Henry
Mar 11 at 9:58
$begingroup$
Your second method needs to consider $sqrt20k-3$ too as that might balance the $sqrt5$ and make the overall result rational - in fact it does not, but you have not shown it does not
$endgroup$
– Henry
Mar 11 at 9:58
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
If $25$ divides $n^2+3n+6,$ the later will be divisible by $5$
$n^2+3n+6equiv n^2-2n+1pmod5$
So, we need $nequiv1pmod5implies n=1+5m$
$(5m+1)^2+3(5m+1)+6=25m^2+25m+10notequiv0pmod25$
$endgroup$
$begingroup$
@Dietrich, extra dot
$endgroup$
– lab bhattacharjee
Mar 11 at 13:13
add a comment |
$begingroup$
Hint: We have
$$
n^2+3n+6=(n+4)^2 bmod 5
$$
Now look at Bill's answer at this duplicate:
$n^2 + 3n +5$ is not divisible by $121$
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
If $25$ divides $n^2+3n+6,$ the later will be divisible by $5$
$n^2+3n+6equiv n^2-2n+1pmod5$
So, we need $nequiv1pmod5implies n=1+5m$
$(5m+1)^2+3(5m+1)+6=25m^2+25m+10notequiv0pmod25$
$endgroup$
$begingroup$
@Dietrich, extra dot
$endgroup$
– lab bhattacharjee
Mar 11 at 13:13
add a comment |
$begingroup$
If $25$ divides $n^2+3n+6,$ the later will be divisible by $5$
$n^2+3n+6equiv n^2-2n+1pmod5$
So, we need $nequiv1pmod5implies n=1+5m$
$(5m+1)^2+3(5m+1)+6=25m^2+25m+10notequiv0pmod25$
$endgroup$
$begingroup$
@Dietrich, extra dot
$endgroup$
– lab bhattacharjee
Mar 11 at 13:13
add a comment |
$begingroup$
If $25$ divides $n^2+3n+6,$ the later will be divisible by $5$
$n^2+3n+6equiv n^2-2n+1pmod5$
So, we need $nequiv1pmod5implies n=1+5m$
$(5m+1)^2+3(5m+1)+6=25m^2+25m+10notequiv0pmod25$
$endgroup$
If $25$ divides $n^2+3n+6,$ the later will be divisible by $5$
$n^2+3n+6equiv n^2-2n+1pmod5$
So, we need $nequiv1pmod5implies n=1+5m$
$(5m+1)^2+3(5m+1)+6=25m^2+25m+10notequiv0pmod25$
edited Mar 11 at 13:13
answered Mar 11 at 10:08
lab bhattacharjeelab bhattacharjee
226k15158275
226k15158275
$begingroup$
@Dietrich, extra dot
$endgroup$
– lab bhattacharjee
Mar 11 at 13:13
add a comment |
$begingroup$
@Dietrich, extra dot
$endgroup$
– lab bhattacharjee
Mar 11 at 13:13
$begingroup$
@Dietrich, extra dot
$endgroup$
– lab bhattacharjee
Mar 11 at 13:13
$begingroup$
@Dietrich, extra dot
$endgroup$
– lab bhattacharjee
Mar 11 at 13:13
add a comment |
$begingroup$
Hint: We have
$$
n^2+3n+6=(n+4)^2 bmod 5
$$
Now look at Bill's answer at this duplicate:
$n^2 + 3n +5$ is not divisible by $121$
$endgroup$
add a comment |
$begingroup$
Hint: We have
$$
n^2+3n+6=(n+4)^2 bmod 5
$$
Now look at Bill's answer at this duplicate:
$n^2 + 3n +5$ is not divisible by $121$
$endgroup$
add a comment |
$begingroup$
Hint: We have
$$
n^2+3n+6=(n+4)^2 bmod 5
$$
Now look at Bill's answer at this duplicate:
$n^2 + 3n +5$ is not divisible by $121$
$endgroup$
Hint: We have
$$
n^2+3n+6=(n+4)^2 bmod 5
$$
Now look at Bill's answer at this duplicate:
$n^2 + 3n +5$ is not divisible by $121$
answered Mar 11 at 10:05
Dietrich BurdeDietrich Burde
80.6k647104
80.6k647104
add a comment |
add a comment |
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$begingroup$
Your second method needs to consider $sqrt20k-3$ too as that might balance the $sqrt5$ and make the overall result rational - in fact it does not, but you have not shown it does not
$endgroup$
– Henry
Mar 11 at 9:58