Answer: Hello there!
A) both channels start at the same hour; 5:10 am, then we could define this hour as t = 0, where t represents the number of minutes after the 5:10 am
we know that chanel 2 gives the weather every 10 minutes, this is at t = 10, 20, 30, etc.
and channel 4 gives the wheater every 12 minutes, this is t = 12, 24, 36, etc
now, we want to find the t such both channels are given the weather.
Then we need to find the Minimal common multiple between 10 and 12.
you have two ways to doing this:
1) a more intuitive one:
Let's write the sets of time where each channel shows the weather, and let's see the first number that is common to both sets.
channel 2: t = (10, 20, 30, 40, 50, 60, 70, 80, 90,....)
channel 4:: t = (12, 24, 36, 48, 60 , 72,.......)
the first number that is common to both sets is t = 60, this means that both channels give the weather at t = 60, or 60 minutes after the 5:10 am,
this is at 6:10 am
2) Using the minimal common multiple between 12 and 10, that is the product between 12 and 10 divided by the maximal common denominator between 12 and 10 (in this case is 2)
then t = 12*10/2 = 60, we got the same result as before.
b) The expression 48 +40 gives the number of minutes that the two channels spend on weather reports in the early morning. We want to write them as a multiple of a sum whose addends have no common factors.
=(48 + 40)
now we start to put common factors "out" of the parentheses.
= 2(24 + 20)
= 2*2(12 + 10) = 4*(12 + 10)
= 2*4(6+ 5) = 8*(6 + 5)
and we know that 5 and 6 don't have any common factors.