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Britney enjoys bird-watching and observed two types of birds traveling this season: ducks and seagulls. While the ducks traveled in flocks of 1010, the seagulls traveled in flocks of 1212. If Britney observed the same total number of ducks and seagulls, what is the smallest number of ducks that she could have observed?\newline_____ ducks

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Q. Britney enjoys bird-watching and observed two types of birds traveling this season: ducks and seagulls. While the ducks traveled in flocks of 1010, the seagulls traveled in flocks of 1212. If Britney observed the same total number of ducks and seagulls, what is the smallest number of ducks that she could have observed?\newline_____ ducks
  1. Identify Approach: Identify the mathematical approach: Since Britney observed equal numbers of ducks and seagulls, we need to find the Least Common Multiple (LCM) of the flock sizes to determine the smallest number of each bird she could have seen.
  2. Calculate LCM: Calculate the LCM of 1010 (ducks) and 1212 (seagulls): \newline10=2×510 = 2 \times 5\newline12=22×312 = 2^2 \times 3\newlineLCM = 22×3×5=602^2 \times 3 \times 5 = 60\newlineThis means Britney observed at least 6060 birds of each type.
  3. Determine Ducks: Determine the number of ducks: Since ducks travel in flocks of 1010, divide the total number of ducks by the size of each flock: 60 ducks10 ducks per flock=6\frac{60 \text{ ducks}}{10 \text{ ducks per flock}} = 6 flocks of ducks.

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