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Difficulty: MediumFactors, Multiples, and Prime Factorization

A school supply drive has 7272 notebooks, 9696 pens, and 120120 pencils. Volunteers want to pack these supplies into identical boxes so that each box contains the same number of notebooks, the same number of pens, and the same number of pencils, with no supplies left over. If they pack the maximum possible number of boxes, how many pens will be in each box?

  1. A
    24
  2. 4Answer
  3. C
    8
  4. D
    12
  5. E
    1440

Answer

4
To find the number of pens in each box, we must first find the greatest common divisor (GCD) of the three quantities (7272, 9696, and 120120) to determine the maximum number of identical boxes that can be created. The prime factorizations are 72=23×3272 = 2^3 \times 3^2, 96=25×396 = 2^5 \times 3, and 120=23×3×5120 = 2^3 \times 3 \times 5. The GCD is the product of the lowest powers of the common prime factors: 23×3=242^3 \times 3 = 24. This means a maximum of 2424 boxes can be packed. Dividing the 9696 pens by the 2424 boxes gives 44 pens per box.

Step-by-Step Solution

1
Find the prime factorization of each of the three quantities: 72, 96, and 120.
72=23×3272 = 2^3 \times 3^2, 96=25×396 = 2^5 \times 3, and 120=23×3×5120 = 2^3 \times 3 \times 5.
To find the greatest common divisor, we first need to break down each number into its prime factors.
2
Determine the Greatest Common Divisor (GCD) of 72, 96, and 120 by taking the lowest power of each common prime factor.
The common prime factors are 2 and 3. The lowest power of 2 is 232^3, and the lowest power of 3 is 313^1. Thus, GCD(72,96,120)=23×3=8×3=24\text{GCD}(72, 96, 120) = 2^3 \times 3 = 8 \times 3 = 24.
The GCD represents the maximum number of identical boxes that can be packed with no leftover items.
3
Divide the total number of pens by the number of boxes to find the number of pens per box.
96÷24=496 \div 24 = 4.
Since there are 96 pens distributed equally among 24 boxes, dividing the total pens by the number of boxes gives the amount in each box.

Key Concept

Using the Greatest Common Divisor (GCD) to solve distribution word problems.
Estimated Time:1m 15s
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Topics

Factors, Multiples, and Prime Factorization

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