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January 25th, 2014, 02:12 AM   #1
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Two Lighthouses. When do they disappear from view together?

Two lighthouses can be seen from the sea. Both lights turn on and off in regular repeating patterns. One is 7 secs on and 16 secs off. The other is 8 secs on and 23 secs off.
15secs ago they became visible at precisely the same moment.

a) In how many seconds will they become visible together again?

b) In how many seconds will they disappear from view together?

__________________________________________________ ______________
a) is easy. Lighthouse 1) 7+16=23sec cycle
Lighthouse 2) 8+23=31sec cycle
Apply LCM 23*31=713 sec. They will become visible at the same moment every 713 sec.
Subtract when they were visible at the same moment for the last time: 713-15sec=698sec.
They will become visible together again in 698sec.

b) This is actually a different problem
IS THERE AN MATHEMATICAL APPROACH?
Making a drawing the answer is 628-15=613sec. But I want to know whether there's a way to do it mathematically.
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January 25th, 2014, 05:50 AM   #2
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Re: Two Lighthouses. When do they disappear from view togeth

If they both become visible 15 seconds ago, one "disappeared" 15- 7= 8 seconds ago and the other 15- 8= 7 seconds ago. The two cycles have, as you say, periods 23 and 31 seconds. So the first "disappears" at 23n- 8 seconds for some integer n and the other at 31m- 7 seconds for some integer m. You want to solve the "Diophantine equation" 23n- 8= 31m- 7 which is the same as 23n- 31m= 1. (A "Diophantine equation" is an equation in which the solutions must be integers.)

Here is how I would solve that: 23 divides into 31 once with remainder 8. That is, 31- 23= 8. 8 divides into 23 twice with remainder 7. That is, 23- 2(= 7. 7 divides into 8 once with remainder 1. That is, 8- 7= 1.

Replace the 7 in that last equation with 23- 2(: 8- (23- 2()= 3(- 23= 1. Replace that 8 with 31- 23: 3(31- 23)- 23= 3(31)- 4(23)= 23(-4)- 31(-3)= 1. That tells us that one solution is n= -4 and m= -3.

But it is easy to see that m= -3+ 23i and n= -4+ 31i is also a solution for any integer i: 31(-3+ 23i)- 23(-4+ 31i)= -93+ (31)(23i)- (-92)- (23)(31i)= 1 for all i. We need to take i= 1, so that m= 20 and n= 27 is the smallest positive solution. After 27 cycles, the first light will disappear in 23(27)- 8= 613 seconds and, after 20 cycles, the second light will disappear in 31(20)- 7= 613 seconds.
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January 25th, 2014, 01:02 PM   #3
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Re: Two Lighthouses. When do they disappear from view togeth

Thank you! very helpful indeed!

Just one thing:

Is there a step-by-step method to solve the diophantine equation or you just see how you could solve it without a method?

I refer to this part:
"Here is how I would solve that: 23 divides into 31 once with remainder 8. That is, 31- 23= 8. 8 divides into 23 twice with remainder 7. That is, 23- 2(= 7. 7 divides into 8 once with remainder 1. That is, 8- 7= 1.

Replace the 7 in that last equation with 23- 2(: 8- (23- 2()= 3(- 23= 1. Replace that 8 with 31- 23: 3(31- 23)- 23= 3(31)- 4(23)= 23(-4)- 31(-3)= 1. That tells us that one solution is n= -4 and m= -3."

Quote:
Originally Posted by HallsofIvy
If they both become visible 15 seconds ago, one "disappeared" 15- 7= 8 seconds ago and the other 15- 8= 7 seconds ago. The two cycles have, as you say, periods 23 and 31 seconds. So the first "disappears" at 23n- 8 seconds for some integer n and the other at 31m- 7 seconds for some integer m. You want to solve the "Diophantine equation" 23n- 8= 31m- 7 which is the same as 23n- 31m= 1. (A "Diophantine equation" is an equation in which the solutions must be integers.)

Here is how I would solve that: 23 divides into 31 once with remainder 8. That is, 31- 23= 8. 8 divides into 23 twice with remainder 7. That is, 23- 2(= 7. 7 divides into 8 once with remainder 1. That is, 8- 7= 1.

Replace the 7 in that last equation with 23- 2(: 8- (23- 2()= 3(- 23= 1. Replace that 8 with 31- 23: 3(31- 23)- 23= 3(31)- 4(23)= 23(-4)- 31(-3)= 1. That tells us that one solution is n= -4 and m= -3.

But it is easy to see that m= -3+ 23i and n= -4+ 31i is also a solution for any integer i: 31(-3+ 23i)- 23(-4+ 31i)= -93+ (31)(23i)- (-92)- (23)(31i)= 1 for all i. We need to take i= 1, so that m= 20 and n= 27 is the smallest positive solution. After 27 cycles, the first light will disappear in 23(27)- 8= 613 seconds and, after 20 cycles, the second light will disappear in 31(20)- 7= 613 seconds.
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January 25th, 2014, 02:08 PM   #4
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Re: Two Lighthouses. When do they disappear from view togeth

23n - 31m = 1
31(n - m) - 8n = 1
31 = (1 + 8n)/(n - m)
31 is a factor of 1 + 8n
31 * 1 - 1 = 30
31 * 3 - 1 = 92
31 * 5 - 1 = 154
31 * 7 - 1 = 216. 216/8 = 27. n = 27. (1 + 8 * 27)/31 = 7. n - m = 7. m = 20.
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January 25th, 2014, 04:27 PM   #5
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Re: Two Lighthouses. When do they disappear from view togeth

Thanks greg1313. I see it for this example. But I am trying with a different set of numbers to see if I can make it work on a different example, and can't. I give the details:

DATA:
A) 5'' on 17''off
B) 13''on 23" off
15" started to shine at the same time.
______________________________________________
A) 5+17= 22 sec cycle
B) 13+28= 41 sec cycle

A)15-5----> -10 #15"ago was on for 5'' (then off)
B)15-13--> -2 #15'' ago was on for 13" (then off)

Diphantine Equation
22n-10=41m-2; 22n-41m=8

41(n-m)-19n=8
41=(8+19n)/(n-m)
41 is a factor of 8+19n
41*1-8 = 33 /19=1,7
41*3-8 = 115/19=6,05
41*5-8 = 197/19=10,36
(...)
41*47-8=1919/19=101 -->integer

So n=101.
22(101)-41m=8; m=-1026/41 = 25,02 (not an integer :S)

Drawing a diagram, I know the answer for the question: "when will they disappear from view together the next time?" is 423 secs after they became visible at the same moment, but I can't prove it mathematically... :S

not even using a diphantine solver I get this result http://www.math.uwaterloo.ca/~snburris/ ... inear.html

Am I doing something wrong?

Quote:
Originally Posted by greg1313
23n - 31m = 1
31(n - m) - 8n = 1
31 = (1 + 8n)/(n - m)
31 is a factor of 1 + 8n
31 * 1 - 1 = 30
31 * 3 - 1 = 92
31 * 5 - 1 = 154
31 * 7 - 1 = 216. 216/8 = 27. n = 27. (1 + 8 * 27)/31 = 7. n - m = 7. m = 20.
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January 25th, 2014, 04:52 PM   #6
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Re: Two Lighthouses. When do they disappear from view togeth

41 * 9 - 8 = 361, n = 19, m = 19 - 9 = 10.
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January 26th, 2014, 01:32 AM   #7
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Re: Two Lighthouses. When do they disappear from view togeth

Thank you!
hence I do not need to use prime numbers for the multiplications, but any number?

Quote:
Originally Posted by greg1313
41 * 9 - 8 = 361, n = 19, m = 19 - 9 = 10.
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January 27th, 2014, 02:41 PM   #8
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Quote:
Originally Posted by kuaiat
Is there a step-by-step method . . .
Yes, and the steps have been demonstrated.
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