two cyclists go around a race track repeatedly. the first cyclist takes 105 seconds to complete a circuit. the other cyclist takes 75 seconds. they begin together at the starting line.

calculate the length of time, in minutes and seconds that would have elapsed before they are together again at the starting line.

To calculate the time it takes for both cyclists to be together at the starting line again, we need to find the least common multiple (LCM) of the two given times. The LCM is the smallest number that is divisible by both 105 and 75.

To find the LCM, we can use the prime factorization method or the division method. Let's use the division method:

Step 1: Divide both numbers by their common factors, starting with the smallest prime number, 2:
105 ÷ 2 = 52.5
75 ÷ 2 = 37.5

Step 2: Divide the resulting numbers by 2 again (if possible):
52.5 ÷ 2 = 26.25
37.5 ÷ 2 = 18.75

Since we cannot divide both numbers by 2 anymore, we move to the next prime number, which is 3.

Step 3: Divide the remaining numbers by 3:
26.25 ÷ 3 = 8.75
18.75 ÷ 3 = 6.25

Next, we divide by 5 since it is a prime number.

Step 4: Divide the remaining numbers by 5:
8.75 ÷ 5 = 1.75
6.25 ÷ 5 = 1.25

After dividing by 5, we can see that both numbers can be expressed as integers. Thus, we can stop dividing.

Now, we multiply all the divisors together to find the LCM:
2 × 2 × 3 × 5 × 1.75 = 105

Therefore, it takes 105 seconds for both cyclists to be together again at the starting line.

To convert this to minutes and seconds, we divide 105 seconds by 60:
105 ÷ 60 = 1.75

We obtain a decimal value of 1.75 minutes, which is equivalent to 1 minute and 0.75 minutes. To convert the decimal part to seconds, we multiply it by 60:
0.75 × 60 = 45

Hence, the time it takes for both cyclists to be together again at the starting line is 1 minute and 45 seconds.