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Chapter 2: Problem 67

Find the GCF. $$17,31,81$$

Short Answer

Expert verified
The GCF of 17, 31, and 81 is 1.

Step by step solution

01

Understand The Concept of GCF

The GCF (Greatest Common Factor) of several integers is the largest positive integer that divides each of the integers without leaving a remainder.
02

Identify the Prime Numbers

Here we observe that 17 and 31 are both prime numbers. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A prime is divisible only 1 and by itself.
03

Check Division

We now check whether 81 can be divided by either 17 or 31. If it can't, it means that the GCF is 1 because the only common factor of any pair of prime numbers is 1. Since 81 isn't a multiple of either 17 or 31, we can conclude that the GCF of 17, 31, and 81 is 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Numbers
Prime numbers play a significant role in determining the Greatest Common Factor (GCF). They possess unique characteristics:
  • A prime number is only divisible by 1 and itself.
  • Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, and 31.
  • Prime numbers are the building blocks of numbers, as every integer can be factored into a product of primes.
In the context of the problem, both 17 and 31 are identified as prime numbers. When we calculate the GCF with primes, one quick insight is that if the third number, such as 81, isn't divisible by any of the primes in question, the GCF is likely to be 1, because the only common divisor for a set of primes with a non-multiple is 1.
Division
Division is an essential mathematical operation for finding the GCF. It helps to check which numbers are factors of others.
  • When dividing, you are essentially distributing one number into equal parts of another.
  • For example, checking if 81 can be divided exactly by 17 or 31 means seeing if 81 contains these numbers as factors.
  • If a number cannot be divided evenly without a remainder, it signals an absence of factors.
In this context, by dividing 81 by 17 and 31, we confirm that neither is a factor of 81. Hence, since 17 and 31 are primes, when they are not factors of 81, their only common divisor is 1, reinforcing that the GCF is 1.
Factors
Understanding factors is at the heart of finding the GCF. Factors are numbers that divide into another number without leaving a remainder.
  • Every number has at least two factors: 1 and itself.
  • Prime numbers have no other factors, making them unique.
  • The GCF emerges from identifying common factors among numbers.
In the given exercise, while examining 17, 31, and 81, we recognize the importance of factors. Since 17 and 31 are primes, they only share a common factor with 81 when 81 can be divided by these prime numbers. In this scenario, since 81 is not divisible by either, their only shared factor is 1, resulting in a GCF of 1.

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