To determine the least amount of plates and cups Tim will need to buy to have equal amounts of each, we need to find the least common multiple (LCM) of the quantities in each package.
The plates come in packs of 24, and the cups come in packs of 16.
To find the LCM of 24 and 16, we can use the prime factorization method:
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Prime factorization:
- For 24: \( 24 = 2^3 \times 3^1 \)
- For 16: \( 16 = 2^4 \)
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LCM: The LCM takes the highest power of each prime factor from both factorizations:
- The highest power of 2 is \( 2^4 \) (from 16).
- The highest power of 3 is \( 3^1 \) (from 24).
So, \[ \text{LCM} = 2^4 \times 3^1 = 16 \times 3 = 48 \]
Thus, the least common multiple of 24 and 16 is 48.
Now, to find out how many packs Tim needs to buy:
- For plates: \( \frac{48}{24} = 2 \) packs of plates
- For cups: \( \frac{48}{16} = 3 \) packs of cups
Tim will buy 2 packs of plates (which equals \( 2 \times 24 = 48 \) plates) and 3 packs of cups (which equals \( 3 \times 16 = 48 \) cups).
Therefore, the least amount of plates and cups that Tim will have to buy to get the same number is 48.
The answer is: 48