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If x, y, z are positive integers satisfying 13x = 17y = 19z, find the minimum value of x + y + z.

 Jan 6, 2020
 #1
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Hint: Find the LCM of 13, 17, and 19

 Jan 6, 2020
 #2
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Maybe find HCF of these values then find LCM.

 

from.

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 Jan 6, 2020
 #3
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Let's find the LCM of them all

\(13(13,26,39,52,65,78,91,104,117,130,143,156,169,182,195,208,221,234,247,260,273,286,299,312,325,338,351,364,377,390,403,416,429,442,455,468,481,494,507,520,533,546,559,572,585,598,611,624,637)\)

\(17(17,34,51,68,85,102,119,136,153,170,187,204,221,238,255,272,289,306,323,340,357,374,391,408,425,442,459,476,493,510,527,544,561,578,595,612,629,646,663,680,697,714,731,748,765,782,799,816,833) \)

\(19(19,38,57,76,95,114,133,152,171,190,209,228,247,266,285,304,323,342,361,380,399,418,437,456,475,494,513,532,551,570,589,608,627,646,665,684,703,722,741,760,779,798,817,836,855,874,893,912,931)\)

There is no LCM for them at least what one can write in an exam......... 

 Jan 6, 2020
 #4
avatar+29198 
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Note that 13, 17 and 19 are all primes.

 

Let x = 17*19, y = 13*19, z = 13*17

 

Then x + y + z = 791

 Jan 6, 2020

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