I have 2 watches with a 12 hour cucle. One gains 2 minutes a day and the other loses 3 minutes a day. If I set them at the correst time,houw many days will it be before they next together tell the correct time?
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We have two twelve-hour watches that are initially set to the correct time. The first watch gains two minutes per day, while the second watch loses three minutes per day. We need to find after how many days both watches will show the correct time again simultaneously.
To solve this problem, we need to understand when a twelve-hour watch shows the correct time again. A twelve-hour watch completes one full cycle every twelve hours, which equals seven hundred twenty minutes. The watch will show the correct time again when its cumulative error equals exactly seven hundred twenty minutes, or any multiple of seven hundred twenty.
Now let's calculate when each watch individually shows the correct time again. Watch one gains two minutes per day, so it needs seven hundred twenty divided by two equals three hundred sixty days to gain a full seven hundred twenty minutes. Watch two loses three minutes per day, so it needs seven hundred twenty divided by three equals two hundred forty days to lose seven hundred twenty minutes. Watch one shows correct time every three hundred sixty days, while watch two shows correct time every two hundred forty days.
To find when both watches show the correct time simultaneously, we need to calculate the least common multiple of three hundred sixty and two hundred forty days. Using prime factorization, three hundred sixty equals two to the third power times three squared times five. Two hundred forty equals two to the fourth power times three times five. The LCM takes the highest power of each prime factor: two to the fourth power times three squared times five, which equals sixteen times nine times five, giving us seven hundred twenty days.
To summarize our solution: A twelve-hour watch shows the correct time again when its cumulative error equals seven hundred twenty minutes. Watch one gains seven hundred twenty minutes in three hundred sixty days, while watch two loses seven hundred twenty minutes in two hundred forty days. The least common multiple of three hundred sixty and two hundred forty is seven hundred twenty. Therefore, both watches will show the correct time simultaneously after exactly seven hundred twenty days.