Explanation:
1.
we want to cut (= split or divide) the pieces into equal strips of max. width.
so, we are looking for the largest integer number that is a factor for both, 72 and 90.
that is the GCF (greatest common factor).
let's do a prime factor analysis (starting with 2):
72 :
72 / 2 = 36
36 / 2 = 18
18 / 2 = 9
9 / 2 no, next one
9 / 3 = 3
3 / 3 = 1 finished
72 = 2³×3²
90 :
90 / 2 = 45
45 / 2 no, next one
45 / 3 = 15
15 / 3 = 5
5 / 3 no, next one
5 / 5 = 1 finished
90 = 2×3²×5
the GCF is the combination of the longest streak per prime factor they share.
that is
2 and 3²
so, the GCF = 2×3² = 18
she can cut them all into strips of 18 in width.
2.
this is how often we have to add 12 to itself (multiply 12 by a factor), and how often we have to add 8 to itself so that both results are equal.
the smallest common result is the LCM (the least common multiple).
we get this again via prime factor analysis.
12 :
12 / 2 = 6
6 / 2 = 3
3 / 2 no, next one
3 / 3 = 1 finished
12 = 2²×3
8 :
8 / 2 = 4
4 / 2 = 2
2 / 2 = 1 finished
8 = 2³
the LCM is the combination of the longest streaks of the prime factors.
that is
2³ and 3
so, the LCM = 2³×3 = 24 days
in 24 days they will exercise together again.
3.
the same principle as 2.
how often do we need to add the numbers to themselves until we reach the same result. we need the LCM.
prime factor analysis :
8 :
we know already
8 = 2³
18 :
18 / 2 = 9
9 / 2 no, next one
9 / 3 = 3
3 / 3 = 1 finished
18 = 2×3²
the LCM is the combination of the longest streaks of the prime factors.
that is 2³ and 3²
the LCM = 2³×3² = 72 in
Cean needs 72/8 = 9 pieces.
Ruth needs 72/18 = 4 pieces.