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Difficulty: MediumFactors, Multiples, and Primes

A botanist has 120120 tulip bulbs, 168168 daffodil bulbs, and 216216 crocus bulbs. She wants to plant them in separate rows such that each row contains only one type of flower, and every row has the exact same number of bulbs. What is the greatest number of bulbs she can plant in a single row?

Answer: 24 bulbs

Answer

24
The greatest number of bulbs per row must perfectly divide the total count of each flower type without remainders. Therefore, we calculate the Greatest Common Factor (GCF) of 120, 168, and 216. The prime factorizations reveal the common factors are 232^3 and 313^1, which multiply to 24.

Step-by-Step Solution

1
Identify the mathematical operation required by the scenario.
The problem asks for the greatest number that divides 120, 168, and 216 evenly. This means finding the Greatest Common Factor (GCF).
Each row must have the same number of bulbs and only one flower type, meaning the row size must be a common divisor of all three total quantities.
2
Find the prime factorization of 120, 168, and 216.
120=23×3×5120 = 2^3 \times 3 \times 5, 168=23×3×7168 = 2^3 \times 3 \times 7, and 216=23×33216 = 2^3 \times 3^3.
Prime factorization is a systematic method to determine the common factors of multiple large numbers.
3
Identify the common prime factors and select their lowest powers.
The common prime factors are 2 and 3. Their lowest powers across the three numbers are 232^3 and 313^1.
The GCF is formed by multiplying the lowest powers of all prime factors that appear in every number's factorization.
4
Calculate the GCF.
23×3=8×3=242^3 \times 3 = 8 \times 3 = 24.
Multiplying the selected prime factors yields the greatest possible number of bulbs per row.

Key Concept

Greatest Common Factor (GCF)
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Topics

Factors, Multiples, and Primes

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