1. Write the multisets (bags) of prime factors of given numbers. i. 160 ii. 120 iii. 250 2. Write the multiplicities of each element of multisets (bags) in Part 2-1(i,ii,iii) separately. 3. Determine the cardinalities of each multiset (bag) in Part 2-1(i,ii,iii).

Solution:

$i) 160=2^5\times 5^1$

$ii) 120=2^3 \times 3^1\times 5^1$

$iii) 250=2^1\times5^3$

1) So, Multiset of 160 of prime factors = $\{2, 2,2,2,2,5\}$

Multiset of 120 of prime factors = $\{2, 2,2,3, 5\}$

Multiset of 250 of prime factors = $\{2, 5,5,5\}$

2)

i) 2 has multiplicity 5,

5 has multiplicity 1

ii) 2 has multiplicity 3,

3 has multiplicity 1,

5 has multiplicity 1

iii) 2 has multiplicity 1

5 has multiplicity 3

3) $i) A=\{2,2,2,2,2,5\}, \;\; |A|=6$

$ii) A=\{2,2,2,3,5\},\;\; |A|=5$

$iii) A=\{2,5,5,5\},\;\; |A|=4$