Final answer:
To determine how many times two watches tick together in the first hour, calculate the LCM of their ticking intervals. The LCM of 95 and 323 seconds is 1615 seconds. In one hour, the watches tick together 3 times, which does not match any of the provided options. Therefore, it's possible there is a mistake in the options provided or an error in the assumptions used while calculating the answer.
Step-by-step explanation:
To calculate how many times two watches will tick together in the first hour, we need to find the Least Common Multiple (LCM) of the two watch ticking intervals. The first watch ticks every 95 seconds, and the second tick every 323 seconds. To find their LCM, we'll need to factor each number and then find the smallest number that can be divided by both 95 and 323.
The prime factors of 95 are 5 and 19, and for 323, these are 17 and 19. Therefore, the LCM of 95 and 323 is 5 x 17 x 19 = 1615 seconds. This number is the interval at which both watches tick together.
Now, we need to determine how many of these intervals occur in one hour (3600 seconds). We divide 3600 by 1615, which gives us approximately 2.23. Since we can't have a part of a ticking event, we round down to 2 full intervals within one hour.
However, both watches tick together at the start as well, so we must add 1 to this count. Thus, the watches tick together a total of 2 + 1 = 3 times in the first hour.
None of the provided options (a) 100 times, (b) 101 times, (c) 99 times, (d) 102 times match the calculated answer. Therefore, it's possible there is a mistake in the options provided or an error in the assumptions used while calculating the answer.