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Question

Difficulty: MediumFactors, Multiples, and Prime Factorization

For a positive integer nn, the greatest common factor of nn and 120120 is 1515, and the least common multiple of nn and 120120 is 360360. What is the value of nn?

  1. A
    5
  2. B
    24
  3. 45Answer
  4. D
    90
  5. E
    180

Answer

45
The correct answer is 45. The relationship between two positive integers aa and bb and their GCF and LCM is a×b=GCF(a,b)×LCM(a,b)a \times b = \text{GCF}(a, b) \times \text{LCM}(a, b). Substituting a=na = n, b=120b = 120, GCF=15\text{GCF} = 15, and LCM=360\text{LCM} = 360 gives n×120=15×360n \times 120 = 15 \times 360. Dividing both sides of the equation by 120 yields n=45n = 45.

Step-by-Step Solution

1
Identify the given values and the mathematical relationship between two positive integers, their greatest common factor (GCF), and their least common multiple (LCM).
Given a=na = n, b=120b = 120, GCF(n,120)=15\text{GCF}(n, 120) = 15, and LCM(n,120)=360\text{LCM}(n, 120) = 360. The relationship is a×b=GCF(a,b)×LCM(a,b)a \times b = \text{GCF}(a, b) \times \text{LCM}(a, b).
This formula provides a direct algebraic path to solve for the unknown integer when the other parameters are known.
2
Substitute the given values into the relationship formula.
n×120=15×360n \times 120 = 15 \times 360
To set up an equation containing only the unknown variable nn.
3
Isolate and solve for nn.
n=15×360120=15×3=45n = \frac{15 \times 360}{120} = 15 \times 3 = 45
To find the specific integer value of nn.

Key Concept

The product of two positive integers is equal to the product of their greatest common factor (GCF) and their least common multiple (LCM).

Alternative Method

Another way to solve this is to write the prime factorizations. Since 120=23×3×5120 = 2^3 \times 3 \times 5, and their GCF is 15=3×515 = 3 \times 5, nn must contain 3×53 \times 5 but cannot contain any factor of 2. Since their LCM is 360=23×32×5360 = 2^3 \times 3^2 \times 5, nn must contain 323^2 because 120 only has 313^1. Thus, the prime factorization of nn is 32×5=453^2 \times 5 = 45.
Estimated Time:1m 0s
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Topics

Factors, Multiples, and Prime Factorization

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