Find of two arithmetic progressions meet
I was working a problem to find if two AP meet or not and came across the below post
Formula to find if two AP meet
and in that the formula given by user is
$$A_n=A_1+(n−1)d$$
$$B_m=B_1+(m−1)D$$
and if $A_n=B_m$ then they meet. But somehow it failed for my case of $A_1=14,d=4 B_1=98,D=2$.
Could you please tell me where i went wrong or the formula was wrong?
sequences-and-series arithmetic arithmetic-progressions
edited 3 hours ago
user26857
39.2k123882
asked 3 hours ago
hardyz009
12
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hardyz009 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
I was working a problem to find if two AP meet or not and came across the below post
Formula to find if two AP meet
and in that the formula given by user is
$$A_n=A_1+(n−1)d$$
$$B_m=B_1+(m−1)D$$
and if $A_n=B_m$ then they meet. But somehow it failed for my case of $A_1=14,d=4 B_1=98,D=2$.
Could you please tell me where i went wrong or the formula was wrong?
sequences-and-series arithmetic arithmetic-progressions
edited 3 hours ago
user26857
39.2k123882
asked 3 hours ago
hardyz009
12
New contributor
hardyz009 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Welcome to Math SE. By definition, the formula is right. In your failure case, what are the values of $n$ for $A_n$ and $m$ for $B_m$ that you tried using?
– John Omielan
3 hours ago
so as per the explanation given by user i got n as 2 and m as 1.
– hardyz009
3 hours ago
Thanks for providing those 2 values. Based on your provided values, the $2$ arithmetic progressions don't meet at that point, as you determined yourself.
– John Omielan
3 hours ago
as per the formula they dont but the actual answer is they do meet and i am trying to figure out where did i go wrong
– hardyz009
3 hours ago
they both meet at 162.
– hardyz009
3 hours ago
add a comment |
I was working a problem to find if two AP meet or not and came across the below post
Formula to find if two AP meet
and in that the formula given by user is
$$A_n=A_1+(n−1)d$$
$$B_m=B_1+(m−1)D$$
and if $A_n=B_m$ then they meet. But somehow it failed for my case of $A_1=14,d=4 B_1=98,D=2$.
Could you please tell me where i went wrong or the formula was wrong?
sequences-and-series arithmetic arithmetic-progressions
edited 3 hours ago
user26857
39.2k123882
asked 3 hours ago
hardyz009
12
New contributor
hardyz009 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I was working a problem to find if two AP meet or not and came across the below post
Formula to find if two AP meet
and in that the formula given by user is
$$A_n=A_1+(n−1)d$$
$$B_m=B_1+(m−1)D$$
and if $A_n=B_m$ then they meet. But somehow it failed for my case of $A_1=14,d=4 B_1=98,D=2$.
Could you please tell me where i went wrong or the formula was wrong?
sequences-and-series arithmetic arithmetic-progressions
sequences-and-series arithmetic arithmetic-progressions
edited 3 hours ago
user26857
39.2k123882
asked 3 hours ago
hardyz009
12
New contributor
hardyz009 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 3 hours ago
user26857
39.2k123882
asked 3 hours ago
hardyz009
12
New contributor
hardyz009 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 3 hours ago
user26857
39.2k123882
edited 3 hours ago
user26857
39.2k123882
edited 3 hours ago
user26857
39.2k123882
39.2k123882
asked 3 hours ago
hardyz009
12
New contributor
hardyz009 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
hardyz009
12
asked 3 hours ago
hardyz009
12
12
New contributor
hardyz009 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
hardyz009 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
hardyz009 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Welcome to Math SE. By definition, the formula is right. In your failure case, what are the values of $n$ for $A_n$ and $m$ for $B_m$ that you tried using?
– John Omielan
3 hours ago
so as per the explanation given by user i got n as 2 and m as 1.
– hardyz009
3 hours ago
Thanks for providing those 2 values. Based on your provided values, the $2$ arithmetic progressions don't meet at that point, as you determined yourself.
– John Omielan
3 hours ago
as per the formula they dont but the actual answer is they do meet and i am trying to figure out where did i go wrong
– hardyz009
3 hours ago
they both meet at 162.
– hardyz009
3 hours ago
add a comment |
Welcome to Math SE. By definition, the formula is right. In your failure case, what are the values of $n$ for $A_n$ and $m$ for $B_m$ that you tried using?
– John Omielan
3 hours ago
so as per the explanation given by user i got n as 2 and m as 1.
– hardyz009
3 hours ago
Thanks for providing those 2 values. Based on your provided values, the $2$ arithmetic progressions don't meet at that point, as you determined yourself.
– John Omielan
3 hours ago
as per the formula they dont but the actual answer is they do meet and i am trying to figure out where did i go wrong
– hardyz009
3 hours ago
they both meet at 162.
– hardyz009
3 hours ago
Welcome to Math SE. By definition, the formula is right. In your failure case, what are the values of $n$ for $A_n$ and $m$ for $B_m$ that you tried using?
– John Omielan
3 hours ago
Welcome to Math SE. By definition, the formula is right. In your failure case, what are the values of $n$ for $A_n$ and $m$ for $B_m$ that you tried using?
– John Omielan
3 hours ago
so as per the explanation given by user i got n as 2 and m as 1.
– hardyz009
3 hours ago
so as per the explanation given by user i got n as 2 and m as 1.
– hardyz009
3 hours ago
Thanks for providing those 2 values. Based on your provided values, the $2$ arithmetic progressions don't meet at that point, as you determined yourself.
– John Omielan
3 hours ago
Thanks for providing those 2 values. Based on your provided values, the $2$ arithmetic progressions don't meet at that point, as you determined yourself.
– John Omielan
3 hours ago
as per the formula they dont but the actual answer is they do meet and i am trying to figure out where did i go wrong
– hardyz009
3 hours ago
as per the formula they dont but the actual answer is they do meet and i am trying to figure out where did i go wrong
– hardyz009
3 hours ago
they both meet at 162.
– hardyz009
3 hours ago
they both meet at 162.
– hardyz009
3 hours ago
add a comment |
1 Answer
1
active
oldest
votes
The $2$ sequences meet, but with $n$ being $22$, not $2$, for $m$ being $1$, with them meeting at $98$, as can be easily confirmed by plugging the values into the equations:
$$14 + left(22 - 1right)4 = 14 + left(21right)4 = 14 + 84 = 98$$
$$98 + left(1 - 1right)2 = 98 + 0 = 98$$
Note to determine $n$ and $m$, subtracting the second general formula from the first general one provided in the question gives:
$$84 = left(n - 1right)4 - left(m - 1right)2$$
$$84 = 4n - 4 - 2m + 2$$
$$86 = 4n - 2m$$
$$43 = 2n - m$$
Since both $m$ and $n$ must be $geq 1$, so this limits what there values can be. As $43$ is odd and $2n$ is even, the values of $m$ must all be odd. The smallest odd value is $1$, with this giving $2n = 43 + 1 = 44$, so $n = 22$.
Since $84$ is a multiple of $4$, the values which work is that $n - 1 = 84/4 = 21$, so $n = 22$, with this giving $m = 1$. For each increase in $n$, the right side increases by $2$, so $m$ must increase by $2$ (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n,mright)$ values which work are $left(22,1right),left(23,3right),left(24,5right),ldots$, with this going up to $left(38,33right)$ which gives $162$, as stated by the question's author, hardyz009, in his/her comment to the original question.
answered 3 hours ago
John Omielan
3317
how did u compute value of n and m.Because as per the explanation we compute n and m by dividing d and D with the gcd. Hence n=4/2 and m=2/2
– hardyz009
3 hours ago
$hardz009, I have added a few more details to my answer, so I hope it explains it better for you and that you now understand it.
– John Omielan
3 hours ago
when you get the equation 84=(m−1)2−(n−1)4 but to calculate we just consider 9n-1)=84 shouldnt the above equation be resolved to 2n-m=41, because we just cant break the equation in between to find values of n and m.
– hardyz009
2 hours ago
@hardz009 Yes, the equation can be simplified to $43 = 2n - m$. However, note that $n$ and $m$ need to each be $ge 1$, so this limits what values they can take. Since 43 is an odd number and $2n$ is even, this means that $m$ must be odd, with the 1st value being 1. This gives that $n = 22$. For each increase in $n$, the right side increases by 2, so $m$ must increase by 2 (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n, mright)$ values which work are $left(22, 1right), left(23, 3right), left(24, 5right), ldots$.
– John Omielan
2 hours ago
but then there has to be a way of calculating n and m with 43=2n-m the point you stated is valid but if it would have been a bigger number or lhs had been even then we wouldnt be doing a hit and trial to compute n and m.
– hardyz009
2 hours ago
|
show 3 more comments
Your Answer
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1 Answer
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The $2$ sequences meet, but with $n$ being $22$, not $2$, for $m$ being $1$, with them meeting at $98$, as can be easily confirmed by plugging the values into the equations:
$$14 + left(22 - 1right)4 = 14 + left(21right)4 = 14 + 84 = 98$$
$$98 + left(1 - 1right)2 = 98 + 0 = 98$$
Note to determine $n$ and $m$, subtracting the second general formula from the first general one provided in the question gives:
$$84 = left(n - 1right)4 - left(m - 1right)2$$
$$84 = 4n - 4 - 2m + 2$$
$$86 = 4n - 2m$$
$$43 = 2n - m$$
Since both $m$ and $n$ must be $geq 1$, so this limits what there values can be. As $43$ is odd and $2n$ is even, the values of $m$ must all be odd. The smallest odd value is $1$, with this giving $2n = 43 + 1 = 44$, so $n = 22$.
Since $84$ is a multiple of $4$, the values which work is that $n - 1 = 84/4 = 21$, so $n = 22$, with this giving $m = 1$. For each increase in $n$, the right side increases by $2$, so $m$ must increase by $2$ (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n,mright)$ values which work are $left(22,1right),left(23,3right),left(24,5right),ldots$, with this going up to $left(38,33right)$ which gives $162$, as stated by the question's author, hardyz009, in his/her comment to the original question.
answered 3 hours ago
John Omielan
3317
how did u compute value of n and m.Because as per the explanation we compute n and m by dividing d and D with the gcd. Hence n=4/2 and m=2/2
– hardyz009
3 hours ago
$hardz009, I have added a few more details to my answer, so I hope it explains it better for you and that you now understand it.
– John Omielan
3 hours ago
when you get the equation 84=(m−1)2−(n−1)4 but to calculate we just consider 9n-1)=84 shouldnt the above equation be resolved to 2n-m=41, because we just cant break the equation in between to find values of n and m.
– hardyz009
2 hours ago
@hardz009 Yes, the equation can be simplified to $43 = 2n - m$. However, note that $n$ and $m$ need to each be $ge 1$, so this limits what values they can take. Since 43 is an odd number and $2n$ is even, this means that $m$ must be odd, with the 1st value being 1. This gives that $n = 22$. For each increase in $n$, the right side increases by 2, so $m$ must increase by 2 (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n, mright)$ values which work are $left(22, 1right), left(23, 3right), left(24, 5right), ldots$.
– John Omielan
2 hours ago
but then there has to be a way of calculating n and m with 43=2n-m the point you stated is valid but if it would have been a bigger number or lhs had been even then we wouldnt be doing a hit and trial to compute n and m.
– hardyz009
2 hours ago
|
show 3 more comments
The $2$ sequences meet, but with $n$ being $22$, not $2$, for $m$ being $1$, with them meeting at $98$, as can be easily confirmed by plugging the values into the equations:
$$14 + left(22 - 1right)4 = 14 + left(21right)4 = 14 + 84 = 98$$
$$98 + left(1 - 1right)2 = 98 + 0 = 98$$
Note to determine $n$ and $m$, subtracting the second general formula from the first general one provided in the question gives:
$$84 = left(n - 1right)4 - left(m - 1right)2$$
$$84 = 4n - 4 - 2m + 2$$
$$86 = 4n - 2m$$
$$43 = 2n - m$$
Since both $m$ and $n$ must be $geq 1$, so this limits what there values can be. As $43$ is odd and $2n$ is even, the values of $m$ must all be odd. The smallest odd value is $1$, with this giving $2n = 43 + 1 = 44$, so $n = 22$.
Since $84$ is a multiple of $4$, the values which work is that $n - 1 = 84/4 = 21$, so $n = 22$, with this giving $m = 1$. For each increase in $n$, the right side increases by $2$, so $m$ must increase by $2$ (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n,mright)$ values which work are $left(22,1right),left(23,3right),left(24,5right),ldots$, with this going up to $left(38,33right)$ which gives $162$, as stated by the question's author, hardyz009, in his/her comment to the original question.
answered 3 hours ago
John Omielan
3317
how did u compute value of n and m.Because as per the explanation we compute n and m by dividing d and D with the gcd. Hence n=4/2 and m=2/2
– hardyz009
3 hours ago
$hardz009, I have added a few more details to my answer, so I hope it explains it better for you and that you now understand it.
– John Omielan
3 hours ago
when you get the equation 84=(m−1)2−(n−1)4 but to calculate we just consider 9n-1)=84 shouldnt the above equation be resolved to 2n-m=41, because we just cant break the equation in between to find values of n and m.
– hardyz009
2 hours ago
@hardz009 Yes, the equation can be simplified to $43 = 2n - m$. However, note that $n$ and $m$ need to each be $ge 1$, so this limits what values they can take. Since 43 is an odd number and $2n$ is even, this means that $m$ must be odd, with the 1st value being 1. This gives that $n = 22$. For each increase in $n$, the right side increases by 2, so $m$ must increase by 2 (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n, mright)$ values which work are $left(22, 1right), left(23, 3right), left(24, 5right), ldots$.
– John Omielan
2 hours ago
but then there has to be a way of calculating n and m with 43=2n-m the point you stated is valid but if it would have been a bigger number or lhs had been even then we wouldnt be doing a hit and trial to compute n and m.
– hardyz009
2 hours ago
|
show 3 more comments
The $2$ sequences meet, but with $n$ being $22$, not $2$, for $m$ being $1$, with them meeting at $98$, as can be easily confirmed by plugging the values into the equations:
$$14 + left(22 - 1right)4 = 14 + left(21right)4 = 14 + 84 = 98$$
$$98 + left(1 - 1right)2 = 98 + 0 = 98$$
Note to determine $n$ and $m$, subtracting the second general formula from the first general one provided in the question gives:
$$84 = left(n - 1right)4 - left(m - 1right)2$$
$$84 = 4n - 4 - 2m + 2$$
$$86 = 4n - 2m$$
$$43 = 2n - m$$
Since both $m$ and $n$ must be $geq 1$, so this limits what there values can be. As $43$ is odd and $2n$ is even, the values of $m$ must all be odd. The smallest odd value is $1$, with this giving $2n = 43 + 1 = 44$, so $n = 22$.
Since $84$ is a multiple of $4$, the values which work is that $n - 1 = 84/4 = 21$, so $n = 22$, with this giving $m = 1$. For each increase in $n$, the right side increases by $2$, so $m$ must increase by $2$ (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n,mright)$ values which work are $left(22,1right),left(23,3right),left(24,5right),ldots$, with this going up to $left(38,33right)$ which gives $162$, as stated by the question's author, hardyz009, in his/her comment to the original question.
answered 3 hours ago
John Omielan
3317
The $2$ sequences meet, but with $n$ being $22$, not $2$, for $m$ being $1$, with them meeting at $98$, as can be easily confirmed by plugging the values into the equations:
$$14 + left(22 - 1right)4 = 14 + left(21right)4 = 14 + 84 = 98$$
$$98 + left(1 - 1right)2 = 98 + 0 = 98$$
Note to determine $n$ and $m$, subtracting the second general formula from the first general one provided in the question gives:
$$84 = left(n - 1right)4 - left(m - 1right)2$$
$$84 = 4n - 4 - 2m + 2$$
$$86 = 4n - 2m$$
$$43 = 2n - m$$
Since both $m$ and $n$ must be $geq 1$, so this limits what there values can be. As $43$ is odd and $2n$ is even, the values of $m$ must all be odd. The smallest odd value is $1$, with this giving $2n = 43 + 1 = 44$, so $n = 22$.
Since $84$ is a multiple of $4$, the values which work is that $n - 1 = 84/4 = 21$, so $n = 22$, with this giving $m = 1$. For each increase in $n$, the right side increases by $2$, so $m$ must increase by $2$ (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n,mright)$ values which work are $left(22,1right),left(23,3right),left(24,5right),ldots$, with this going up to $left(38,33right)$ which gives $162$, as stated by the question's author, hardyz009, in his/her comment to the original question.
answered 3 hours ago
John Omielan
3317
edited 1 hour ago
answered 3 hours ago
John Omielan
3317
answered 3 hours ago
John Omielan
3317
answered 3 hours ago
John Omielan
3317
3317
how did u compute value of n and m.Because as per the explanation we compute n and m by dividing d and D with the gcd. Hence n=4/2 and m=2/2
– hardyz009
3 hours ago
$hardz009, I have added a few more details to my answer, so I hope it explains it better for you and that you now understand it.
– John Omielan
3 hours ago
when you get the equation 84=(m−1)2−(n−1)4 but to calculate we just consider 9n-1)=84 shouldnt the above equation be resolved to 2n-m=41, because we just cant break the equation in between to find values of n and m.
– hardyz009
2 hours ago
@hardz009 Yes, the equation can be simplified to $43 = 2n - m$. However, note that $n$ and $m$ need to each be $ge 1$, so this limits what values they can take. Since 43 is an odd number and $2n$ is even, this means that $m$ must be odd, with the 1st value being 1. This gives that $n = 22$. For each increase in $n$, the right side increases by 2, so $m$ must increase by 2 (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n, mright)$ values which work are $left(22, 1right), left(23, 3right), left(24, 5right), ldots$.
– John Omielan
2 hours ago
but then there has to be a way of calculating n and m with 43=2n-m the point you stated is valid but if it would have been a bigger number or lhs had been even then we wouldnt be doing a hit and trial to compute n and m.
– hardyz009
2 hours ago
|
show 3 more comments
how did u compute value of n and m.Because as per the explanation we compute n and m by dividing d and D with the gcd. Hence n=4/2 and m=2/2
– hardyz009
3 hours ago
$hardz009, I have added a few more details to my answer, so I hope it explains it better for you and that you now understand it.
– John Omielan
3 hours ago
when you get the equation 84=(m−1)2−(n−1)4 but to calculate we just consider 9n-1)=84 shouldnt the above equation be resolved to 2n-m=41, because we just cant break the equation in between to find values of n and m.
– hardyz009
2 hours ago
@hardz009 Yes, the equation can be simplified to $43 = 2n - m$. However, note that $n$ and $m$ need to each be $ge 1$, so this limits what values they can take. Since 43 is an odd number and $2n$ is even, this means that $m$ must be odd, with the 1st value being 1. This gives that $n = 22$. For each increase in $n$, the right side increases by 2, so $m$ must increase by 2 (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n, mright)$ values which work are $left(22, 1right), left(23, 3right), left(24, 5right), ldots$.
– John Omielan
2 hours ago
but then there has to be a way of calculating n and m with 43=2n-m the point you stated is valid but if it would have been a bigger number or lhs had been even then we wouldnt be doing a hit and trial to compute n and m.
– hardyz009
2 hours ago
how did u compute value of n and m.Because as per the explanation we compute n and m by dividing d and D with the gcd. Hence n=4/2 and m=2/2
– hardyz009
3 hours ago
how did u compute value of n and m.Because as per the explanation we compute n and m by dividing d and D with the gcd. Hence n=4/2 and m=2/2
– hardyz009
3 hours ago
$hardz009, I have added a few more details to my answer, so I hope it explains it better for you and that you now understand it.
– John Omielan
3 hours ago
$hardz009, I have added a few more details to my answer, so I hope it explains it better for you and that you now understand it.
– John Omielan
3 hours ago
when you get the equation 84=(m−1)2−(n−1)4 but to calculate we just consider 9n-1)=84 shouldnt the above equation be resolved to 2n-m=41, because we just cant break the equation in between to find values of n and m.
– hardyz009
2 hours ago
when you get the equation 84=(m−1)2−(n−1)4 but to calculate we just consider 9n-1)=84 shouldnt the above equation be resolved to 2n-m=41, because we just cant break the equation in between to find values of n and m.
– hardyz009
2 hours ago
@hardz009 Yes, the equation can be simplified to $43 = 2n - m$. However, note that $n$ and $m$ need to each be $ge 1$, so this limits what values they can take. Since 43 is an odd number and $2n$ is even, this means that $m$ must be odd, with the 1st value being 1. This gives that $n = 22$. For each increase in $n$, the right side increases by 2, so $m$ must increase by 2 (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n, mright)$ values which work are $left(22, 1right), left(23, 3right), left(24, 5right), ldots$.
– John Omielan
2 hours ago
@hardz009 Yes, the equation can be simplified to $43 = 2n - m$. However, note that $n$ and $m$ need to each be $ge 1$, so this limits what values they can take. Since 43 is an odd number and $2n$ is even, this means that $m$ must be odd, with the 1st value being 1. This gives that $n = 22$. For each increase in $n$, the right side increases by 2, so $m$ must increase by 2 (as it is being subtracted) to compensate to keep it an equation. Thus, the sets of $left(n, mright)$ values which work are $left(22, 1right), left(23, 3right), left(24, 5right), ldots$.
– John Omielan
2 hours ago
but then there has to be a way of calculating n and m with 43=2n-m the point you stated is valid but if it would have been a bigger number or lhs had been even then we wouldnt be doing a hit and trial to compute n and m.
– hardyz009
2 hours ago
but then there has to be a way of calculating n and m with 43=2n-m the point you stated is valid but if it would have been a bigger number or lhs had been even then we wouldnt be doing a hit and trial to compute n and m.
– hardyz009
2 hours ago
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hardyz009 is a new contributor. Be nice, and check out our Code of Conduct.
hardyz009 is a new contributor. Be nice, and check out our Code of Conduct.
hardyz009 is a new contributor. Be nice, and check out our Code of Conduct.
hardyz009 is a new contributor. Be nice, and check out our Code of Conduct.
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Welcome to Math SE. By definition, the formula is right. In your failure case, what are the values of $n$ for $A_n$ and $m$ for $B_m$ that you tried using?
– John Omielan
3 hours ago
so as per the explanation given by user i got n as 2 and m as 1.
– hardyz009
3 hours ago
Thanks for providing those 2 values. Based on your provided values, the $2$ arithmetic progressions don't meet at that point, as you determined yourself.
– John Omielan
3 hours ago
as per the formula they dont but the actual answer is they do meet and i am trying to figure out where did i go wrong
– hardyz009
3 hours ago
they both meet at 162.
– hardyz009
3 hours ago