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Question

Difficulty: MediumFactors, Multiples, and Primes

A logistics coordinator is packing identical emergency relief kits. They have 180180 bottles of water, 252252 energy bars, and 324324 bandages. If every kit must contain the exact same combination of supplies with no items left over, what is the greatest number of kits that can be assembled?

  1. A
    18
  2. 36Answer
  3. C
    72
  4. D
    11,340

Answer

The greatest number of kits that can be assembled is 36.
The scenario describes dividing three sets of items into identical groups with no remainder, which means finding the Greatest Common Factor (GCF). Breaking down the numbers into prime factors yields: 180=22×32×5180 = 2^2 \times 3^2 \times 5, 252=22×32×7252 = 2^2 \times 3^2 \times 7, and 324=22×34324 = 2^2 \times 3^4. The factors shared by all three are 222^2 and 323^2. Multiplying these together (4×94 \times 9) gives the GCF of 3636. Therefore, a maximum of 3636 kits can be made, each containing exactly 55 water bottles, 77 energy bars, and 99 bandages.

Step-by-Step Solution

1
Identify the mathematical operation required by the scenario.
We need to find the Greatest Common Factor (GCF) of 180, 252, and 324.
The problem asks for the maximum number of identical kits that can be made by dividing all supplies without any leftovers, which aligns with the definition of the GCF.
2
Determine the prime factorization for each quantity.
180=22×32×5180 = 2^2 \times 3^2 \times 5
252=22×32×7252 = 2^2 \times 3^2 \times 7
324=22×34324 = 2^2 \times 3^4
Prime factorization is a reliable method to find all common prime factors among multiple large numbers.
3
Identify the common prime factors and their lowest exponents.
The common prime bases are 2 and 3. Their lowest exponents across all factorizations are 222^2 and 323^2.
The GCF is the product of the lowest power of each prime factor that appears in all of the numbers.
4
Calculate the final GCF by multiplying the common prime factors.
22×32=4×9=362^2 \times 3^2 = 4 \times 9 = 36
Multiplying the lowest common prime factors gives the greatest numerical value that divides evenly into all three supply quantities.

Key Concept

Greatest Common Factor (GCF) via Prime Factorization

Alternative Method

Instead of prime factorization, you can check the factors of the common difference. The difference between the quantities is 7272 (252−180=72252 - 180 = 72 and 324−252=72324 - 252 = 72). The factors of 7272 are 72,36,24,18,12,9,8,6,4,3,2,172, 36, 24, 18, 12, 9, 8, 6, 4, 3, 2, 1. Checking these against 180180 from largest to smallest, 7272 does not divide 180180, but 3636 does (180÷36=5180 \div 36 = 5). Since 3636 divides all three numbers, it is the GCF.
Estimated Time:1m 30s
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Topics

Factors, Multiples, and Primes

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