Explanation:
I guess the question is how many packages of forks and how many packages of spoons did she need to buy at least to have an equal number of forks and spoons ?
we need to find the LCM (the Least Common Multiple) of 32 and 24.
for this we need to split both numbers into their prime factors (prime numbers that divide the number without remainder).
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1 finished
32 = 2×2×2×2×2 = 2⁵
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
3 ÷ 2 no, next prime number
3 ÷ 3 = 1 finished
24 = 2×2×2×3
the LCM is the product of the longest chains per prime factor in both numbers. that is
2×2×2×2×2 × 3 = 32×3 = 96
to get 96 forks she needs to buy 3 packages of forks (32 × 3 = 96).
to get 96 spoons she needs to buy 4 packages of spoons (24 × 4 = 96).
if she needs more that 96 sets, she needs to buy corresponding multiples of 96 items (multiples of 3 packages of forks, and fitting multiples of 4 packages of spoons to get multiples of 96 utensils per type) to keep the ratio of 1:1 forks and spoons.
e.g.
2×96 = 192 forks and 192 spoons.
that means then
2×3 = 6 packages of forks
2×4 = 8 packages of spoons
3× 96 = 288 forks and 288 spoons.
that means then
3×3 = 9 packages of forks
3×4 = 12 packages of spoons
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