One computer in a lab is programmed to back up data at the turn of the minute every five minutes. Another computer is programmed to back up data at the turn of the minute every two minutes. Find the number of times in twenty-four hours that the two computers back up data at the same time.
(Assume that the computers do not back up at the start of the 24-hour period.)
To find the number of times the two computers back up data at the same time in 24 hours, we need to find the least common multiple (LCM) of 5 and 2, which is 10.
Since 24 hours is equivalent to 1440 minutes, we can divide 1440 by 10 to find the number of times the two computers back up data at the same time:
1440 / 10 = 144
Therefore, the two computers back up data at the same time 144 times in 24 hours.
A whole number larger than two leaves a remainder of two when divided by each of the numbers three, four, five, and six. What is the smallest such number?
To find the smallest number that leaves a remainder of 2 when divided by 3, 4, 5, and 6, we need to find the least common multiple (LCM) of 3, 4, 5, and 6 and add 2 to get the smallest number that satisfies the given conditions.
The LCM of 3, 4, 5, and 6 is 60.
Therefore, the smallest number that satisfies the conditions is:
60 + 2 = 62
So, the smallest number larger than 2 that leaves a remainder of 2 when divided by 3, 4, 5, and 6 is 62.
What is the least common multiple of 51, 68 and 85?
To find the least common multiple (LCM) of 51, 68, and 85, we can first find the prime factorization of each number:
51 = 3 * 17
68 = 2^2 * 17
85 = 5 * 17
Then, the LCM is the product of all the prime factors, including the highest power of each prime that appears in the factorization of any number:
LCM(51, 68, 85) = 2^2 * 3 * 5 * 17 = 1020
Therefore, the least common multiple of 51, 68, and 85 is 1020.